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Mirrors > Home > MPE Home > Th. List > fnmap | Structured version Visualization version GIF version |
Description: Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
fnmap | ⊢ ↑m Fn (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-map 8763 | . 2 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
2 | mapex 8767 | . . 3 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V) | |
3 | 2 | el2v 3451 | . 2 ⊢ {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V |
4 | 1, 3 | fnmpoi 7998 | 1 ⊢ ↑m Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 {cab 2713 Vcvv 3443 × cxp 5629 Fn wfn 6488 ⟶wf 6489 ↑m cmap 8761 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-map 8763 |
This theorem is referenced by: elmapex 8782 |
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