Theorem elcnvlem 43625

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) ↦ ⟨(2nd ‘𝑥), (1st ‘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 5857	. 2 ⊢ (𝐴 ∈ ◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2		fveq2 6876	. . . . 5 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3		vex 3463	. . . . . . 7 ⊢ 𝑢 ∈ V
4		vex 3463	. . . . . . 7 ⊢ 𝑣 ∈ V
5	3, 4	opelvv 5694	. . . . . 6 ⊢ ⟨𝑢, 𝑣⟩ ∈ (V × V)
6	3, 4	op2ndd 7999	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑥) = 𝑣)
7	3, 4	op1std 7998	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑥) = 𝑢)
8	6, 7	opeq12d 4857	. . . . . . 7 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd ‘𝑥), (1st ‘𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9		elcnvlem.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd ‘𝑥), (1st ‘𝑥)⟩)
10		opex 5439	. . . . . . 7 ⊢ ⟨𝑣, 𝑢⟩ ∈ V
11	8, 9, 10	fvmpt 6986	. . . . . 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 2819	. . 3 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
15	14	copsex2gb 5785	. 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))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   ↦ cmpt 5201   × cxp 5652  ^◡ccnv 5653  ‘cfv 6531  1^st c1st 7986  2^nd c2nd 7987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-1st 7988  df-2nd 7989

This theorem is referenced by:  elcnvintab  43626