Theorem bj-elid5 37735

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 5814	. . . 4 ⊢ Rel I
2		df-rel 5669	. . . 4 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 233	. . 3 ⊢ I ⊆ (V × V)
4	3	sseli 3941	. 2 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		bj-elid4 37734	. 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
6	4, 5	biadanii 833	1 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913   I cid 5556   × cxp 5660  Rel wrel 5667  ‘cfv 6537  1^st c1st 7984  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7986  df-2nd 7987

This theorem is referenced by: (None)