Theorem ixpfn 8969

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 6660	. 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
2		elixp2 8967	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3	2	simp2bi 1164	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴)
4	1, 3	vtoclga 3540	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   Fn wfn 6561  ‘cfv 6566  Xcixp 8963

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574  df-ixp 8964

This theorem is used by:  ixpprc  8985  undifixp  9000  resixpfo  9002  boxcutc  9007  ixpiunwdom  9621  prdsbasfn  17692  xpsff1o  17789  sscfn1  18042  funcfn2  18094  natfn  18182  pthaus  24012  ptuncnv  24181  ptunhmeo  24182  ptcmplem2  24427  prdsbl  24865  finixpnum  38796  upixp  38931  prdstotbnd  38994  elixpconstg  46377  rrxsnicc  47583  ioorrnopnxrlem  47589  hoidmvlelem3  47880  hspdifhsp  47899  hspmbllem2  47910