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Theorem xpsfrnel 17537

Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 17545. (Contributed by Mario Carneiro, 14-Aug-2015.)

**Assertion**
Ref	Expression
xpsfrnel	⊢ (𝐺 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2_o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝐺

**Proof of Theorem xpsfrnel**
Step	Hyp	Ref	Expression
1		elixp2 8913	. 2 ⊢ (𝐺 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ 𝐺 Fn 2_o ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
2		3ancoma 1096	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐺 Fn 2_o ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2_o ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)))
3		2onn 8656	. . . . . . . . . 10 ⊢ 2_o ∈ ω
4		nnfi 9185	. . . . . . . . . 10 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
5	3, 4	ax-mp 5	. . . . . . . . 9 ⊢ 2_o ∈ Fin
6		fnfi 9199	. . . . . . . . 9 ⊢ ((𝐺 Fn 2_o ∧ 2_o ∈ Fin) → 𝐺 ∈ Fin)
7	5, 6	mpan2 690	. . . . . . . 8 ⊢ (𝐺 Fn 2_o → 𝐺 ∈ Fin)
8	7	elexd 3491	. . . . . . 7 ⊢ (𝐺 Fn 2_o → 𝐺 ∈ V)
9	8	biantrurd 532	. . . . . 6 ⊢ (𝐺 Fn 2_o → (∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))))
10		df2o3 8488	. . . . . . . 8 ⊢ 2_o = {∅, 1_o}
11	10	raleqi 3319	. . . . . . 7 ⊢ (∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ∀𝑘 ∈ {∅, 1_o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))
12		0ex 5301	. . . . . . . 8 ⊢ ∅ ∈ V
13		1oex 8490	. . . . . . . 8 ⊢ 1_o ∈ V
14		fveq2 6891	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐺‘𝑘) = (𝐺‘∅))
15		iftrue 4530	. . . . . . . . 9 ⊢ (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
16	14, 15	eleq12d 2823	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘∅) ∈ 𝐴))
17		fveq2 6891	. . . . . . . . 9 ⊢ (𝑘 = 1_o → (𝐺‘𝑘) = (𝐺‘1_o))
18		1n0 8502	. . . . . . . . . . 11 ⊢ 1_o ≠ ∅
19		neeq1 2999	. . . . . . . . . . 11 ⊢ (𝑘 = 1_o → (𝑘 ≠ ∅ ↔ 1_o ≠ ∅))
20	18, 19	mpbiri 258	. . . . . . . . . 10 ⊢ (𝑘 = 1_o → 𝑘 ≠ ∅)
21		ifnefalse 4536	. . . . . . . . . 10 ⊢ (𝑘 ≠ ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
22	20, 21	syl 17	. . . . . . . . 9 ⊢ (𝑘 = 1_o → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
23	17, 22	eleq12d 2823	. . . . . . . 8 ⊢ (𝑘 = 1_o → ((𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺‘1_o) ∈ 𝐵))
24	12, 13, 16, 23	ralpr 4700	. . . . . . 7 ⊢ (∀𝑘 ∈ {∅, 1_o} (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵))
25	11, 24	bitri 275	. . . . . 6 ⊢ (∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵))
26	9, 25	bitr3di 286	. . . . 5 ⊢ (𝐺 Fn 2_o → ((𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵)))
27	26	pm5.32i 574	. . . 4 ⊢ ((𝐺 Fn 2_o ∧ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))) ↔ (𝐺 Fn 2_o ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵)))
28		3anass 1093	. . . 4 ⊢ ((𝐺 Fn 2_o ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2_o ∧ (𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵))))
29		3anass 1093	. . . 4 ⊢ ((𝐺 Fn 2_o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵) ↔ (𝐺 Fn 2_o ∧ ((𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵)))
30	27, 28, 29	3bitr4i 303	. . 3 ⊢ ((𝐺 Fn 2_o ∧ 𝐺 ∈ V ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2_o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵))
31	2, 30	bitri 275	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐺 Fn 2_o ∧ ∀𝑘 ∈ 2_o (𝐺‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐵)) ↔ (𝐺 Fn 2_o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵))
32	1, 31	bitri 275	1 ⊢ (𝐺 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐺 Fn 2_o ∧ (𝐺‘∅) ∈ 𝐴 ∧ (𝐺‘1_o) ∈ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 Vcvv 3470 ∅c0 4318 ifcif 4524 {cpr 4626 Fn wfn 6537 ‘cfv 6542 ωcom 7864 1_oc1o 8473 2_oc2o 8474 Xcixp 8909 Fincfn 8957

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 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 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-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1o 8480 df-2o 8481 df-ixp 8910 df-en 8958 df-fin 8961

This theorem is referenced by: xpsfrnel2 17539 xpsff1o 17542

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