Theorem ixpfn 8841

Description: A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.)

Assertion

Ref	Expression
ixpfn	⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem ixpfn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq1 6576	. 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
2		elixp2 8839	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3	2	simp2bi 1152	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴)
4	1, 3	vtoclga 3520	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   Fn wfn 6480  ‘cfv 6485  Xcixp 8835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-ixp 8836

This theorem is referenced by:  ixpprc  8857  undifixp  8872  resixpfo  8874  boxcutc  8879  ixpiunwdom  9495  prdsbasfn  17425  xpsff1o  17522  sscfn1  17775  funcfn2  17827  natfn  17915  pthaus  23621  ptuncnv  23790  ptunhmeo  23791  ptcmplem2  24036  prdsbl  24474  finixpnum  37972  upixp  38096  prdstotbnd  38161  elixpconstg  45536  rrxsnicc  46743  ioorrnopnxrlem  46749  hoidmvlelem3  47040  hspdifhsp  47059  hspmbllem2  47070