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Theorem bj-elid2 33423
 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 33422 . . 3 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
21simprbi 484 . 2 (𝐴 ∈ I → (1st𝐴) = (2nd𝐴))
3 xpss 5265 . . . 4 (𝑉 × 𝑊) ⊆ (V × V)
43sseli 3748 . . 3 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
51simplbi2 488 . . 3 (𝐴 ∈ (V × V) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
64, 5syl 17 . 2 (𝐴 ∈ (𝑉 × 𝑊) → ((1st𝐴) = (2nd𝐴) → 𝐴 ∈ I ))
72, 6impbid2 216 1 (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  Vcvv 3351   I cid 5156   × cxp 5247  ‘cfv 6031  1st c1st 7313  2nd c2nd 7314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-2nd 7316 This theorem is referenced by:  bj-eldiag  33428
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