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Mirrors > Home > MPE Home > Th. List > fnpm | Structured version Visualization version GIF version |
Description: Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
fnpm | ⊢ ↑pm Fn (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pm 8097 | . 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | |
2 | vex 3387 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 3387 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | xpex 7195 | . . . 4 ⊢ (𝑦 × 𝑥) ∈ V |
5 | 4 | pwex 5049 | . . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V |
6 | 5 | rabex 5006 | . 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V |
7 | 1, 6 | fnmpt2i 7474 | 1 ⊢ ↑pm Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: {crab 3092 Vcvv 3384 𝒫 cpw 4348 × cxp 5309 Fun wfun 6094 Fn wfn 6095 ↑pm cpm 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-id 5219 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-fv 6108 df-oprab 6881 df-mpt2 6882 df-1st 7400 df-2nd 7401 df-pm 8097 |
This theorem is referenced by: elpmi 8113 pmresg 8122 pmsspw 8129 |
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