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Theorem bj-elid2 33584
Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid2 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Proof of Theorem bj-elid2
StepHypRef Expression
1 bj-elid 33583 . . 3 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
21simprbi 491 . 2 (𝐴 ∈ I → (1st𝐴) = (2nd𝐴))
3 xpss 5328 . . . 4 (𝑉 × 𝑊) ⊆ (V × V)
43sseli 3794 . . 3 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
51simplbi2 495 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
64, 5syl 17 . 2 (𝐴 ∈ (𝑉 × 𝑊) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
72, 6impbid2 218 1 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  Vcvv 3385   I cid 5219   × cxp 5310  cfv 6101  1st c1st 7399  2nd c2nd 7400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-1st 7401  df-2nd 7402
This theorem is referenced by:  bj-eldiag  33590
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