Theorem elinlem 37904

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 3796	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2		fvi 6255	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ( I ‘𝐴) = 𝐴)
3	2	eqcomd 2628	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 = ( I ‘𝐴))
4	3	eleq1d 2686	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ↔ ( I ‘𝐴) ∈ 𝐶))
5	4	pm5.32i 669	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
6	1, 5	bitri 264	1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1990   ∩ cin 3573   I cid 5023  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  elcnvcnvlem  37905