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Theorem elinlem 36923
 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 3758 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
2 fvi 6165 . . . . 5 (𝐴𝐵 → ( I ‘𝐴) = 𝐴)
32eqcomd 2616 . . . 4 (𝐴𝐵𝐴 = ( I ‘𝐴))
43eleq1d 2672 . . 3 (𝐴𝐵 → (𝐴𝐶 ↔ ( I ‘𝐴) ∈ 𝐶))
54pm5.32i 667 . 2 ((𝐴𝐵𝐴𝐶) ↔ (𝐴𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
61, 5bitri 263 1 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ∩ cin 3539   I cid 4948  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  elcnvcnvlem  36924
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