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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 7077 | . . 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 2105 ◡ccnv 5687 “ cima 5691 Fn wfn 6557 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: fpwwe2lem8 10675 rhmpreimaidl 21304 evlslem3 22121 rhmpreimaprmidl 33458 ply1degltel 33594 ply1degleel 33595 ply1degltlss 33596 ply1degltdimlem 33649 ply1degltdim 33650 dimkerim 33654 lvecendof1f1o 33660 zndvdchrrhm 41952 smfsuplem1 46766 |
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