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Theorem elinlem 42184
Description: Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.)
Assertion
Ref Expression
elinlem (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))

Proof of Theorem elinlem
StepHypRef Expression
1 elin 3961 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
2 fvi 6954 . . . . 5 (𝐴𝐵 → ( I ‘𝐴) = 𝐴)
32eqcomd 2738 . . . 4 (𝐴𝐵𝐴 = ( I ‘𝐴))
43eleq1d 2818 . . 3 (𝐴𝐵 → (𝐴𝐶 ↔ ( I ‘𝐴) ∈ 𝐶))
54pm5.32i 575 . 2 ((𝐴𝐵𝐴𝐶) ↔ (𝐴𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
61, 5bitri 274 1 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wcel 2106  cin 3944   I cid 5567  cfv 6533
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541
This theorem is referenced by:  elcnvcnvlem  42185
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