Theorem bj-elid2 33086

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 33085	. . 3 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
2	1	simprbi 480	. 2 ⊢ (𝐴 ∈ I → (1st ‘𝐴) = (2nd ‘𝐴))
3		xpss 5226	. . . 4 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
4	3	sseli 3599	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
5	1	simplbi2 655	. . 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 216	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  Vcvv 3200   I cid 5023   × cxp 5112  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  bj-eldiag  33091