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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 8858 | . 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | |
2 | vex 3466 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | vex 3466 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 2, 3 | xpex 7761 | . . . 4 ⊢ (𝑦 × 𝑥) ∈ V |
5 | 4 | pwex 5384 | . . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V |
6 | 5 | rabex 5339 | . 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V |
7 | 1, 6 | fnmpoi 8084 | 1 ⊢ ↑pm Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: {crab 3419 Vcvv 3462 𝒫 cpw 4607 × cxp 5680 Fun wfun 6548 Fn wfn 6549 ↑pm cpm 8856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-pm 8858 |
This theorem is referenced by: elpmi 8875 pmresg 8899 pmsspw 8906 |
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