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Mirrors > Home > ILE Home > Th. List > fnmap | 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 6597 | . 2 ⊢ ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
2 | vex 2715 | . . 3 ⊢ 𝑦 ∈ V | |
3 | vex 2715 | . . 3 ⊢ 𝑥 ∈ V | |
4 | mapex 6601 | . . 3 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V) | |
5 | 2, 3, 4 | mp2an 423 | . 2 ⊢ {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V |
6 | 1, 5 | fnmpoi 6154 | 1 ⊢ ↑𝑚 Fn (V × V) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 {cab 2143 Vcvv 2712 × cxp 4586 Fn wfn 5167 ⟶wf 5168 ↑𝑚 cmap 6595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-fv 5180 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-map 6597 |
This theorem is referenced by: mapsnen 6758 map1 6759 mapen 6793 mapdom1g 6794 mapxpen 6795 xpmapenlem 6796 hashfacen 10718 omctfn 12242 cnfval 12664 cnpfval 12665 cnpval 12668 ismet 12814 isxmet 12815 xmetunirn 12828 |
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