dfoprab3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dfoprab3		Structured version Visualization version GIF version

Theorem dfoprab3 7998

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 7997	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^st ‘𝑤) / 𝑥][(2^nd ‘𝑤) / 𝑦]𝜓)}
2		fvex 6847	. . . . 5 ⊢ (1^st ‘𝑤) ∈ V
3		fvex 6847	. . . . 5 ⊢ (2^nd ‘𝑤) ∈ V
4		eqcom 2743	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) ↔ (1^st ‘𝑤) = 𝑥)
5		eqcom 2743	. . . . . . . . . 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 7969	. . . . . . . . 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 3817	. . . 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 5165	. 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 2760	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 1541 ∈ wcel 2113 Vcvv 3440 [wsbc 3740 ⟨cop 4586 {copab 5160 × cxp 5622 ‘cfv 6492 {coprab 7359 1^st c1st 7931 2^nd c2nd 7932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-oprab 7362 df-1st 7933 df-2nd 7934

This theorem is referenced by: dfoprab4 7999 cnvoprab 8004 df1st2 8040 df2nd2 8041

W3C validator