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Mirrors > Home > ILE Home > Th. List > fnpm | 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 6678 | . 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | |
2 | vex 2755 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 2755 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | xpex 4759 | . . . 4 ⊢ (𝑦 × 𝑥) ∈ V |
5 | 4 | pwex 4201 | . . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V |
6 | 5 | rabex 4162 | . 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V |
7 | 1, 6 | fnmpoi 6230 | 1 ⊢ ↑pm Fn (V × V) |
Colors of variables: wff set class |
Syntax hints: {crab 2472 Vcvv 2752 𝒫 cpw 3590 × cxp 4642 Fun wfun 5229 Fn wfn 5230 ↑pm cpm 6676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-pm 6678 |
This theorem is referenced by: lmfval 14169 |
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