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Mirrors > Home > MPE Home > Th. List > Mathboxes > elinlem | Structured version Visualization version GIF version |
Description: Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.) |
Ref | Expression |
---|---|
elinlem | ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3964 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
2 | fvi 6967 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ( I ‘𝐴) = 𝐴) | |
3 | 2 | eqcomd 2737 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 = ( I ‘𝐴)) |
4 | 3 | eleq1d 2817 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2105 ∩ cin 3947 I cid 5573 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: elcnvcnvlem 42815 |
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