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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfunsg | Structured version Visualization version GIF version |
Description: Closed form of elfuns 34818. (Contributed by Scott Fenton, 2-May-2014.) |
Ref | Expression |
---|---|
elfunsg | ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Funs ↔ Fun 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2822 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ Funs ↔ 𝐹 ∈ Funs )) | |
2 | funeq 6560 | . 2 ⊢ (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹)) | |
3 | vex 3479 | . . 3 ⊢ 𝑓 ∈ V | |
4 | 3 | elfuns 34818 | . 2 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓) |
5 | 1, 2, 4 | vtoclbg 3558 | 1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ Funs ↔ Fun 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 Fun wfun 6529 Funs cfuns 34740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fo 6541 df-fv 6543 df-1st 7962 df-2nd 7963 df-txp 34757 df-fix 34762 df-funs 34764 |
This theorem is referenced by: (None) |
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