Theorem bj-elid5 34485

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 5691	. . . 4 ⊢ Rel I
2		df-rel 5555	. . . 4 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 232	. . 3 ⊢ I ⊆ (V × V)
4	3	sseli 3956	. 2 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		bj-elid4 34484	. 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
6	4, 5	biadanii 820	1 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3491   ⊆ wss 3929   I cid 5452   × cxp 5546  Rel wrel 5553  ‘cfv 6348  1^st c1st 7680  2^nd c2nd 7681

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-1st 7682  df-2nd 7683

This theorem is referenced by: (None)