Theorem bj-elid5 37135

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 5850	. . . 4 ⊢ Rel I
2		df-rel 5707	. . . 4 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 230	. . 3 ⊢ I ⊆ (V × V)
4	3	sseli 4004	. 2 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		bj-elid4 37134	. 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 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   I cid 5592   × cxp 5698  Rel wrel 5705  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-1st 8030  df-2nd 8031

This theorem is referenced by: (None)