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Theorem dfoprab3 7734

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 7733	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓)}
2		fvex 6658	. . . . 5 ⊢ (1^st ‘𝑤) ∈ V
3		fvex 6658	. . . . 5 ⊢ (2^nd ‘𝑤) ∈ V
4		eqcom 2805	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) ↔ (1^st ‘𝑤) = 𝑥)
5		eqcom 2805	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = 𝑦)
6	4, 5	anbi12i 629	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) ↔ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦))
7		eqopi 7707	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylan2b 596	. . . . . . . 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 226	. . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤))) → (𝜓 ↔ 𝜑))
12	11	ex 416	. . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) → (𝜓 ↔ 𝜑)))
13	2, 3, 12	sbc2iedv 3797	. . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑))
14	13	pm5.32i 578	. . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
15	14	opabbii 5097	. 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 2822	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 ⟨cop 4531 {copab 5092 × cxp 5517 ‘cfv 6324 {coprab 7136 1^st c1st 7669 2^nd c2nd 7670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-oprab 7139 df-1st 7671 df-2nd 7672

This theorem is referenced by: dfoprab4 7735 cnvoprab 7740 df1st2 7776 df2nd2 7777

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