Theorem elcnvlem 43614

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 5888	. 2 ⊢ (𝐴 ∈ ◡𝐵 ↔ ∃𝑢∃𝑣(𝐴 = ⟨𝑢, 𝑣⟩ ∧ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
2		fveq2 6906	. . . . 5 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = (𝐹‘⟨𝑢, 𝑣⟩))
3		vex 3484	. . . . . . 7 ⊢ 𝑢 ∈ V
4		vex 3484	. . . . . . 7 ⊢ 𝑣 ∈ V
5	3, 4	opelvv 5725	. . . . . 6 ⊢ ⟨𝑢, 𝑣⟩ ∈ (V × V)
6	3, 4	op2ndd 8025	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑥) = 𝑣)
7	3, 4	op1std 8024	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑥) = 𝑢)
8	6, 7	opeq12d 4881	. . . . . . 7 ⊢ (𝑥 = ⟨𝑢, 𝑣⟩ → ⟨(2nd ‘𝑥), (1st ‘𝑥)⟩ = ⟨𝑣, 𝑢⟩)
9		elcnvlem.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd ‘𝑥), (1st ‘𝑥)⟩)
10		opex 5469	. . . . . . 7 ⊢ ⟨𝑣, 𝑢⟩ ∈ V
11	8, 9, 10	fvmpt 7016	. . . . . 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 2793	. . . 4 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝐴) = ⟨𝑣, 𝑢⟩)
14	13	eleq1d 2826	. . 3 ⊢ (𝐴 = ⟨𝑢, 𝑣⟩ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ⟨𝑣, 𝑢⟩ ∈ 𝐵))
15	14	copsex2gb 5816	. 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 3480  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  elcnvintab  43615