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Theorem dfoprab3 7989

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 7988	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓)}
2		fvex 6835	. . . . 5 ⊢ (1^st ‘𝑤) ∈ V
3		fvex 6835	. . . . 5 ⊢ (2^nd ‘𝑤) ∈ V
4		eqcom 2736	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) ↔ (1^st ‘𝑤) = 𝑥)
5		eqcom 2736	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = 𝑦)
6	4, 5	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) ↔ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦))
7		eqopi 7960	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylan2b 594	. . . . . . . 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 223	. . . . . 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 3819	. . . 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 5159	. 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 2753	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3436 [wsbc 3742 ⟨cop 4583 {copab 5154 × cxp 5617 ‘cfv 6482 {coprab 7350 1^st c1st 7922 2^nd c2nd 7923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-iota 6438 df-fun 6484 df-fv 6490 df-oprab 7353 df-1st 7924 df-2nd 7925

This theorem is referenced by: dfoprab4 7990 cnvoprab 7995 df1st2 8031 df2nd2 8032

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