elcnvlem - Mathbox for Richard Penner

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Theorem elcnvlem 44115

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 5838	. 2 ⊢ (𝐴 ∈ ^◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2		fveq2 6852	. . . . 5 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3		vex 3448	. . . . . . 7 ⊢ 𝑢 ∈ V
4		vex 3448	. . . . . . 7 ⊢ 𝑣 ∈ V
5	3, 4	opelvv 5676	. . . . . 6 ⊢ ⟨𝑢, 𝑣⟩ ∈ (V × V)
6	3, 4	op2ndd 7966	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (2^nd ‘𝑥) = 𝑣)
7	3, 4	op1std 7965	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (1^st ‘𝑥) = 𝑢)
8	6, 7	opeq12d 4829	. . . . . . 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 5421	. . . . . . 7 ⊢ ⟨𝑣, 𝑢⟩ ∈ V
11	8, 9, 10	fvmpt 6960	. . . . . 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 2803	. . . 4 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = ⟨𝑣, 𝑢⟩)
14	13	eleq1d 2837	. . 3 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
15	14	copsex2gb 5768	. 2 ⊢ (∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵) ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))
16	1, 15	bitri 277	1 ⊢ (𝐴 ∈ ^◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1550 ∃wex 1789 ∈ wcel 2132 Vcvv 3444 ⟨cop 4578 ↦ cmpt 5171 × cxp 5634 ^◡ccnv 5635 ‘cfv 6506 1^st c1st 7953 2^nd c2nd 7954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-iota 6462 df-fun 6508 df-fv 6514 df-1st 7955 df-2nd 7956

This theorem is referenced by: elcnvintab 44116

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