Theorem fveqressseq 7083

Description: If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)

Hypothesis

Ref	Expression
fveqdmss.1	⊢ 𝐷 = dom 𝐵

Assertion

Ref	Expression
fveqressseq	⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷

Proof of Theorem fveqressseq

Step	Hyp	Ref	Expression
1		fveqdmss.1	. . . 4 ⊢ 𝐷 = dom 𝐵
2	1	fveqdmss 7082	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴)
3		dmres 6001	. . . . 5 ⊢ dom (𝐴 ↾ 𝐷) = (𝐷 ∩ dom 𝐴)
4		incom 4197	. . . . . 6 ⊢ (𝐷 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐷)
5		sseqin2 4211	. . . . . . 7 ⊢ (𝐷 ⊆ dom 𝐴 ↔ (dom 𝐴 ∩ 𝐷) = 𝐷)
6	5	biimpi 215	. . . . . 6 ⊢ (𝐷 ⊆ dom 𝐴 → (dom 𝐴 ∩ 𝐷) = 𝐷)
7	4, 6	eqtrid 2780	. . . . 5 ⊢ (𝐷 ⊆ dom 𝐴 → (𝐷 ∩ dom 𝐴) = 𝐷)
8	3, 7	eqtrid 2780	. . . 4 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = 𝐷)
9	8, 1	eqtrdi 2784	. . 3 ⊢ (𝐷 ⊆ dom 𝐴 → dom (𝐴 ↾ 𝐷) = dom 𝐵)
10	2, 9	syl 17	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = dom 𝐵)
11		fvres 6910	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
12	11	adantl 481	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐴‘𝑥))
13		id 22	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) = (𝐵‘𝑥))
14	12, 13	sylan9eq 2788	. . . . . 6 ⊢ ((((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) ∧ (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
15	14	ex 412	. . . . 5 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
16	15	ralimdva 3163	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
17	16	3impia 1115	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
18	2, 7	syl 17	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐷 ∩ dom 𝐴) = 𝐷)
19	3, 18	eqtrid 2780	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → dom (𝐴 ↾ 𝐷) = 𝐷)
20	19	raleqdv 3321	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥) ↔ ∀𝑥 ∈ 𝐷 ((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥)))
21	17, 20	mpbird 257	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))
22		simpll 766	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → Fun 𝐵)
23	1	eleq2i 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ dom 𝐵)
24	23	biimpi 215	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ dom 𝐵)
25	24	adantl 481	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ dom 𝐵)
26		simplr 768	. . . . . . . 8 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ∅ ∉ ran 𝐵)
27		nelrnfvne 7081	. . . . . . . 8 ⊢ ((Fun 𝐵 ∧ 𝑥 ∈ dom 𝐵 ∧ ∅ ∉ ran 𝐵) → (𝐵‘𝑥) ≠ ∅)
28	22, 25, 26, 27	syl3anc 1369	. . . . . . 7 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → (𝐵‘𝑥) ≠ ∅)
29		neeq1 2999	. . . . . . 7 ⊢ ((𝐴‘𝑥) = (𝐵‘𝑥) → ((𝐴‘𝑥) ≠ ∅ ↔ (𝐵‘𝑥) ≠ ∅))
30	28, 29	syl5ibrcom 246	. . . . . 6 ⊢ (((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) = (𝐵‘𝑥) → (𝐴‘𝑥) ≠ ∅))
31	30	ralimdva 3163	. . . . 5 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵) → (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅))
32	31	3impia 1115	. . . 4 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅)
33		fvn0ssdmfun 7078	. . . . 5 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → (𝐷 ⊆ dom 𝐴 ∧ Fun (𝐴 ↾ 𝐷)))
34	33	simprd 495	. . . 4 ⊢ (∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ ∅ → Fun (𝐴 ↾ 𝐷))
35	32, 34	syl 17	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun (𝐴 ↾ 𝐷))
36		simp1 1134	. . 3 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → Fun 𝐵)
37		eqfunfv 7039	. . 3 ⊢ ((Fun (𝐴 ↾ 𝐷) ∧ Fun 𝐵) → ((𝐴 ↾ 𝐷) = 𝐵 ↔ (dom (𝐴 ↾ 𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))))
38	35, 36, 37	syl2anc 583	. 2 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → ((𝐴 ↾ 𝐷) = 𝐵 ↔ (dom (𝐴 ↾ 𝐷) = dom 𝐵 ∧ ∀𝑥 ∈ dom (𝐴 ↾ 𝐷)((𝐴 ↾ 𝐷)‘𝑥) = (𝐵‘𝑥))))
39	10, 21, 38	mpbir2and 712	1 ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2936   ∉ wnel 3042  ∀wral 3057   ∩ cin 3944   ⊆ wss 3945  ∅c0 4318  dom cdm 5672  ran crn 5673   ↾ cres 5674  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550

This theorem is referenced by:  plusfreseq  47220

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