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| Mirrors > Home > MPE Home > Th. List > elpreimad | Structured version Visualization version GIF version | ||
| Description: Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| elpreimad.f | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| elpreimad.b | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
| elpreimad.c | ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶) |
| Ref | Expression |
|---|---|
| elpreimad | ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elpreimad.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
| 2 | elpreimad.c | . 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶) | |
| 3 | elpreimad.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
| 4 | elpreima 7078 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) |
| 6 | 1, 2, 5 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ‘cfv 6561 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fpwwe2lem8 10678 rhmpreimaidl 21287 evlslem3 22104 elrgspnsubrunlem2 33252 rhmpreimaprmidl 33479 ply1degltel 33615 ply1degleel 33616 ply1degltlss 33617 exsslsb 33647 ply1degltdimlem 33673 ply1degltdim 33674 dimkerim 33678 lvecendof1f1o 33684 zndvdchrrhm 41972 smfsuplem1 46826 |
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