Theorem elinlem 43587

Description: Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.)

Assertion

Ref	Expression
elinlem	⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))

Proof of Theorem elinlem

Step	Hyp	Ref	Expression
1		elin 3930	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2		fvi 6937	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ( I ‘𝐴) = 𝐴)
3	2	eqcomd 2735	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 = ( I ‘𝐴))
4	3	eleq1d 2813	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ↔ ( I ‘𝐴) ∈ 𝐶))
5	4	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
6	1, 5	bitri 275	1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ∩ cin 3913   I cid 5532  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519

This theorem is referenced by:  elcnvcnvlem  43588