Theorem bj-elid5 37502

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 5776	. . . 4 ⊢ Rel I
2		df-rel 5632	. . . 4 ⊢ (Rel I ↔ I ⊆ (V × V))
3	1, 2	mpbi 230	. . 3 ⊢ I ⊆ (V × V)
4	3	sseli 3918	. 2 ⊢ (𝐴 ∈ I → 𝐴 ∈ (V × V))
5		bj-elid4 37501	. 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
6	4, 5	biadanii 822	1 ⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890   I cid 5519   × cxp 5623  Rel wrel 5630  ‘cfv 6493  1^st c1st 7934  2^nd c2nd 7935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-1st 7936  df-2nd 7937

This theorem is referenced by: (None)