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| Mirrors > Home > MPE Home > Th. List > fvelimabd | Structured version Visualization version GIF version | ||
| Description: Deduction form of fvelimab 6906. (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 6906 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1541 ∈ wcel 2113 ∃wrex 3060 ⊆ wss 3901 “ cima 5627 Fn wfn 6487 ‘cfv 6492 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-fv 6500 |
| This theorem is referenced by: unima 6909 resf1extb 7876 ghmqusnsglem1 19209 ghmquskerlem1 19212 lmhmima 20999 mdegldg 26027 ig1peu 26136 2ndimaxp 32724 fnpreimac 32749 fsuppcurry1 32803 fsuppcurry2 32804 swrdrn3 33037 esplyfv1 33727 esplyfv 33728 esplyfval3 33730 fnrelpredd 35247 bj-gabima 37141 extoimad 44405 upgrimpths 48155 |
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