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Mirrors > Home > MPE Home > Th. List > fvelimabd | Structured version Visualization version GIF version |
Description: Deduction form of fvelimab 6737. (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 6737 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 “ cima 5558 Fn wfn 6350 ‘cfv 6355 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: unima 6739 lmhmima 19819 mdegldg 24660 ig1peu 24765 fnpreimac 30416 fsuppcurry1 30461 fsuppcurry2 30462 swrdrn3 30629 extoimad 40535 |
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