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Mirrors > Home > MPE Home > Th. List > fvelimabd | Structured version Visualization version GIF version |
Description: Deduction form of fvelimab 6712. (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 6712 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ⊆ wss 3881 “ cima 5522 Fn wfn 6319 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: unima 6714 lmhmima 19812 mdegldg 24667 ig1peu 24772 2ndimaxp 30409 fnpreimac 30434 fsuppcurry1 30487 fsuppcurry2 30488 swrdrn3 30655 fnrelpredd 32472 extoimad 40868 |
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