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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elfunsg | Structured version Visualization version GIF version | ||
| Description: Closed form of elfuns 35916. (Contributed by Scott Fenton, 2-May-2014.) |
| Ref | Expression |
|---|---|
| elfunsg | ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Funs ↔ Fun 𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1 2829 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ Funs ↔ 𝐹 ∈ Funs )) | |
| 2 | funeq 6586 | . 2 ⊢ (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹)) | |
| 3 | vex 3484 | . . 3 ⊢ 𝑓 ∈ V | |
| 4 | 3 | elfuns 35916 | . 2 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓) |
| 5 | 1, 2, 4 | vtoclbg 3557 | 1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Funs ↔ Fun 𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Fun wfun 6555 Funs cfuns 35838 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-fix 35860 df-funs 35862 |
| This theorem is referenced by: (None) |
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