Theorem fnpm 8102

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 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)

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