Theorem fnpm 8775

Description: Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
fnpm	⊢ ↑pm Fn (V × V)

Proof of Theorem fnpm

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pm 8770	. 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2		vex 3445	. . . . 5 ⊢ 𝑦 ∈ V
3		vex 3445	. . . . 5 ⊢ 𝑥 ∈ V
4	2, 3	xpex 7700	. . . 4 ⊢ (𝑦 × 𝑥) ∈ V
5	4	pwex 5326	. . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V
6	5	rabex 5285	. 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
7	1, 6	fnmpoi 8016	1 ⊢ ↑pm Fn (V × V)

Syntax hints:  {crab 3400  Vcvv 3441  𝒫 cpw 4555   × cxp 5623  Fun wfun 6487   Fn wfn 6488   ↑_pm cpm 8768

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-pm 8770

This theorem is referenced by:  elpmi  8787  pmresg  8812  pmsspw  8819