elcnvlem - Mathbox for Richard Penner

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Theorem elcnvlem 43879

Description: Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)

**Hypothesis**
Ref	Expression
elcnvlem.f	⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2^nd ‘𝑥), (1^st ‘𝑥)⟩)

**Assertion**
Ref	Expression
elcnvlem	⊢ (𝐴 ∈ ^◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))

Proof of Theorem elcnvlem Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcnv2 5825	. 2 ⊢ (𝐴 ∈ ^◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2		fveq2 6833	. . . . 5 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3		vex 3443	. . . . . . 7 ⊢ 𝑢 ∈ V
4		vex 3443	. . . . . . 7 ⊢ 𝑣 ∈ V
5	3, 4	opelvv 5663	. . . . . 6 ⊢ ⟨𝑢, 𝑣⟩ ∈ (V × V)
6	3, 4	op2ndd 7944	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (2^nd ‘𝑥) = 𝑣)
7	3, 4	op1std 7943	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (1^st ‘𝑥) = 𝑢)
8	6, 7	opeq12d 4836	. . . . . . 7 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2^nd ‘𝑥), (1^st ‘𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9		elcnvlem.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2^nd ‘𝑥), (1^st ‘𝑥)⟩)
10		opex 5411	. . . . . . 7 ⊢ ⟨𝑣, 𝑢⟩ ∈ V
11	8, 9, 10	fvmpt 6940	. . . . . 6 ⊢ (⟨𝑢, 𝑣⟩ ∈ (V × V) → (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢⟩)
12	5, 11	ax-mp 5	. . . . 5 ⊢ (𝐹‘⟨𝑢, 𝑣⟩) = ⟨𝑣, 𝑢⟩
13	2, 12	eqtrdi 2786	. . . 4 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = ⟨𝑣, 𝑢⟩)
14	13	eleq1d 2820	. . 3 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
15	14	copsex2gb 5754	. 2 ⊢ (∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))
16	1, 15	bitri 275	1 ⊢ (𝐴 ∈ ^◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 Vcvv 3439 ⟨cop 4585 ↦ cmpt 5178 × cxp 5621 ^◡ccnv 5622 ‘cfv 6491 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 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pr 5376 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3399 df-v 3441 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4285 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-iota 6447 df-fun 6493 df-fv 6499 df-1st 7933 df-2nd 7934

This theorem is referenced by: elcnvintab 43880

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