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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 6474 | . 2 ⊢ ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
2 | vex 2644 | . . 3 ⊢ 𝑦 ∈ V | |
3 | vex 2644 | . . 3 ⊢ 𝑥 ∈ V | |
4 | mapex 6478 | . . 3 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V) | |
5 | 2, 3, 4 | mp2an 420 | . 2 ⊢ {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V |
6 | 1, 5 | fnmpoi 6032 | 1 ⊢ ↑𝑚 Fn (V × V) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 {cab 2086 Vcvv 2641 × cxp 4475 Fn wfn 5054 ⟶wf 5055 ↑𝑚 cmap 6472 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-map 6474 |
This theorem is referenced by: mapsnen 6635 map1 6636 mapen 6669 mapdom1g 6670 mapxpen 6671 xpmapenlem 6672 hashfacen 10420 cnfval 12145 cnpfval 12146 cnpval 12148 ismet 12272 isxmet 12273 xmetunirn 12286 |
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