csbopeq1a - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > csbopeq1a		Structured version Visualization version GIF version

Theorem csbopeq1a 7864

Description: Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analogue of csbeq1a 3842). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

**Assertion**
Ref	Expression
csbopeq1a	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)

**Proof of Theorem csbopeq1a**
Step	Hyp	Ref	Expression
1		vex 3426	. . . . 5 ⊢ 𝑥 ∈ V
2		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7815	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝐴) = 𝑦)
4	3	eqcomd 2744	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = (2^nd ‘𝐴))
5		csbeq1a 3842	. . 3 ⊢ (𝑦 = (2^nd ‘𝐴) → 𝐵 = ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐵 = ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
7	1, 2	op1std 7814	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = 𝑥)
8	7	eqcomd 2744	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑥 = (1^st ‘𝐴))
9		csbeq1a 3842	. . 3 ⊢ (𝑥 = (1^st ‘𝐴) → ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
10	8, 9	syl 17	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵)
11	6, 10	eqtr2d 2779	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1^st ‘𝐴) / 𝑥⦌⦋(2^nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ⦋csb 3828 ⟨cop 4564 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fv 6426 df-1st 7804 df-2nd 7805

This theorem is referenced by: dfmpo 7913 f1od2 30958 wdom2d2 40773

W3C validator