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Theorem dfoprab3 7224

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 7223	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓)}
2		fvex 6201	. . . . 5 ⊢ (1^st ‘𝑤) ∈ V
3		fvex 6201	. . . . 5 ⊢ (2^nd ‘𝑤) ∈ V
4		eqcom 2629	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) ↔ (1^st ‘𝑤) = 𝑥)
5		eqcom 2629	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) ↔ (2^nd ‘𝑤) = 𝑦)
6	4, 5	anbi12i 733	. . . . . . . . 9 ⊢ ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) ↔ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦))
7		eqopi 7202	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1^st ‘𝑤) = 𝑥 ∧ (2^nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylan2b 492	. . . . . . . 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 213	. . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤))) → (𝜓 ↔ 𝜑))
12	11	ex 450	. . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1^st ‘𝑤) ∧ 𝑦 = (2^nd ‘𝑤)) → (𝜓 ↔ 𝜑)))
13	2, 3, 12	sbc2iedv 3506	. . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑))
14	13	pm5.32i 669	. . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
15	14	opabbii 4717	. 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 2645	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 ⟨cop 4183 {copab 4712 × cxp 5112 ‘cfv 5888 {coprab 6651 1^st c1st 7166 2^nd c2nd 7167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-oprab 6654 df-1st 7168 df-2nd 7169

This theorem is referenced by: dfoprab4 7225 df1st2 7263 df2nd2 7264

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