elcnvlem - Mathbox for Richard Penner

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Theorem elcnvlem 44125

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 5842	. 2 ⊢ (𝐴 ∈ ^◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2		fveq2 6856	. . . . 5 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3		vex 3452	. . . . . . 7 ⊢ 𝑢 ∈ V
4		vex 3452	. . . . . . 7 ⊢ 𝑣 ∈ V
5	3, 4	opelvv 5680	. . . . . 6 ⊢ ⟨𝑢, 𝑣⟩ ∈ (V × V)
6	3, 4	op2ndd 7970	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (2^nd ‘𝑥) = 𝑣)
7	3, 4	op1std 7969	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (1^st ‘𝑥) = 𝑢)
8	6, 7	opeq12d 4833	. . . . . . 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 5425	. . . . . . 7 ⊢ ⟨𝑣, 𝑢⟩ ∈ V
11	8, 9, 10	fvmpt 6964	. . . . . 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 2807	. . . 4 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = ⟨𝑣, 𝑢⟩)
14	13	eleq1d 2841	. . 3 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
15	14	copsex2gb 5772	. 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 1554 ∃wex 1793 ∈ wcel 2136 Vcvv 3448 ⟨cop 4582 ↦ cmpt 5175 × cxp 5638 ^◡ccnv 5639 ‘cfv 6510 1^st c1st 7957 2^nd c2nd 7958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pr 5384 ax-un 7707

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-rab 3409 df-v 3450 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4475 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5095 df-opab 5157 df-mpt 5176 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-iota 6466 df-fun 6512 df-fv 6518 df-1st 7959 df-2nd 7960

This theorem is referenced by: elcnvintab 44126

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