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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 6820 | . . 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 709 | 1 ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ◡ccnv 5547 “ cima 5551 Fn wfn 6343 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: evlslem3 20221 dimkerim 30922 smfsuplem1 42962 |
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