Theorem elinlem 43556

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 3949	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2		fvi 6966	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ( I ‘𝐴) = 𝐴)
3	2	eqcomd 2740	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 = ( I ‘𝐴))
4	3	eleq1d 2818	. . 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 2107   ∩ cin 3932   I cid 5559  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6495  df-fun 6544  df-fv 6550

This theorem is referenced by:  elcnvcnvlem  43557