![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8392 | . 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | |
2 | vex 3444 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | xpex 7456 | . . . 4 ⊢ (𝑦 × 𝑥) ∈ V |
5 | 4 | pwex 5246 | . . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V |
6 | 5 | rabex 5199 | . 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V |
7 | 1, 6 | fnmpoi 7750 | 1 ⊢ ↑pm Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: {crab 3110 Vcvv 3441 𝒫 cpw 4497 × cxp 5517 Fun wfun 6318 Fn wfn 6319 ↑pm cpm 8390 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-pm 8392 |
This theorem is referenced by: elpmi 8408 pmresg 8417 pmsspw 8424 |
Copyright terms: Public domain | W3C validator |