Theorem sscres 17811

Description: Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
sscres	⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Proof of Theorem sscres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4229	. . 3 ⊢ (𝑆 ∩ 𝑇) ⊆ 𝑆
2		simpl 481	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ (𝑆 ∩ 𝑇))
3	2	elin2d 4199	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ 𝑇)
4		simpr 483	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ (𝑆 ∩ 𝑇))
5	4	elin2d 4199	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ 𝑇)
6	3, 5	ovresd 7592	. . . . 5 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7		eqimss 4038	. . . . 5 ⊢ ((𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
8	6, 7	syl 17	. . . 4 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
9	8	rgen2 3193	. . 3 ⊢ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
10	1, 9	pm3.2i 469	. 2 ⊢ ((𝑆 ∩ 𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
11		simpl 481	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝐻 Fn (𝑆 × 𝑆))
12		inss1 4229	. . . . 5 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13		fnssres 6681	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
14	11, 12, 13	sylancl 584	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15		resres 6000	. . . . . 6 ⊢ ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16		fnresdm 6677	. . . . . . . 8 ⊢ (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
17	16	adantr 479	. . . . . . 7 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
18	17	reseq1d 5986	. . . . . 6 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
19	15, 18	eqtr3id 2781	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20		inxp 5836	. . . . . 6 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇))
21	20	a1i 11	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
22	19, 21	fneq12d 6652	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ↔ (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇))))
23	14, 22	mpbid 231	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
24		simpr 483	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝑆 ∈ 𝑉)
25	23, 11, 24	isssc 17808	. 2 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻 ↔ ((𝑆 ∩ 𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))))
26	10, 25	mpbiri 257	1 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3057   ∩ cin 3946   ⊆ wss 3947   class class class wbr 5150   × cxp 5678   ↾ cres 5682   Fn wfn 6546  (class class class)co 7424   ⊆_cat cssc 17795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-ixp 8921  df-ssc 17798

This theorem is referenced by:  sscid  17812  fullsubc  17841

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