Theorem bj-elid2 32263

Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-elid2	⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))

Proof of Theorem bj-elid2

Step	Hyp	Ref	Expression
1		bj-elid 32262	. . 3 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
2	1	simprbi 479	. 2 ⊢ (𝐴 ∈ I → (1st ‘𝐴) = (2nd ‘𝐴))
3		xpss 5149	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
4	3	sseli 3564	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
5	1	simplbi2 653	. . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = (2nd ‘𝐴) → 𝐴 ∈ I ))
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = (2nd ‘𝐴) → 𝐴 ∈ I ))
7	2, 6	impbid2 215	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173   I cid 4948   × cxp 5036  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by:  bj-eldiag  32268