Theorem elinlem 44175

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 3921	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2		fvi 6944	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ( I ‘𝐴) = 𝐴)
3	2	eqcomd 2769	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 = ( I ‘𝐴))
4	3	eleq1d 2848	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ↔ ( I ‘𝐴) ∈ 𝐶))
5	4	pm5.32i 582	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
6	1, 5	bitri 277	1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2143   ∩ cin 3904   I cid 5542  ‘cfv 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6478  df-fun 6524  df-fv 6530

This theorem is referenced by:  elcnvcnvlem  44176