Theorem bj-elid5 37147

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

Assertion

Ref	Expression
bj-elid5	⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Proof of Theorem bj-elid5

Step	Hyp	Ref	Expression
1		reli 5769	. . . 4 ⊢ Rel I
2		df-rel 5626	. . . 4 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 230	. . 3 ⊢ I ⊆ (V × V)
4	3	sseli 3931	. 2 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		bj-elid4 37146	. 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
6	4, 5	biadanii 821	1 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ⊆ wss 3903   I cid 5513   × cxp 5617  Rel wrel 5624  ‘cfv 6482  1^st c1st 7922  2^nd c2nd 7923

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-1st 7924  df-2nd 7925

This theorem is referenced by: (None)