![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvelimabd | Structured version Visualization version GIF version |
Description: Deduction form of fvelimab 6513. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
fvelimabd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fvelimabd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
fvelimabd | ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelimabd.1 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | fvelimabd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
3 | fvelimab 6513 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) | |
4 | 1, 2, 3 | syl2anc 579 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ⊆ wss 3792 “ cima 5358 Fn wfn 6130 ‘cfv 6135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 |
This theorem is referenced by: wlkresOLD 27021 fnpreimac 30036 extoimad 39420 |
Copyright terms: Public domain | W3C validator |