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Theorem dfoprab3 8039

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

**Hypothesis**
Ref	Expression
dfoprab3.1	⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

**Assertion**
Ref	Expression
dfoprab3	⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Distinct variable groups: 𝑥,𝑦,𝜑 𝜓,𝑤 𝑥,𝑧,𝑤,𝑦 Allowed substitution hints: 𝜑(𝑧,𝑤) 𝜓(𝑥,𝑦,𝑧)

**Proof of Theorem dfoprab3**
Step	Hyp	Ref	Expression
1		dfoprab3s 8038	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓)}
2		fvex 6898	. . . . 5 ⊢ (1^st ‘𝑤) ∈ V
3		fvex 6898	. . . . 5 ⊢ (2^nd ‘𝑤) ∈ V
4		eqcom 2733	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) ↔ (1^st ‘𝑤) = 𝑥)
5		eqcom 2733	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = 𝑦)
6	4, 5	anbi12i 626	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) ↔ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦))
7		eqopi 8010	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylan2b 593	. . . . . . . 8 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤))) → 𝑤 = ⟨𝑥, 𝑦⟩)
9		dfoprab3.1	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤))) → (𝜑 ↔ 𝜓))
11	10	bicomd 222	. . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤))) → (𝜓 ↔ 𝜑))
12	11	ex 412	. . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) → (𝜓 ↔ 𝜑)))
13	2, 3, 12	sbc2iedv 3857	. . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑))
14	13	pm5.32i 574	. . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
15	14	opabbii 5208	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
16	1, 15	eqtr2i 2755	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 [wsbc 3772 ⟨cop 4629 {copab 5203 × cxp 5667 ‘cfv 6537 {coprab 7406 1^st c1st 7972 2^nd c2nd 7973

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fv 6545 df-oprab 7409 df-1st 7974 df-2nd 7975

This theorem is referenced by: dfoprab4 8040 cnvoprab 8045 df1st2 8084 df2nd2 8085

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