Theorem ixpfn 8910

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 6623	. 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
2		elixp2 8908	. . 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 3536	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   Fn wfn 6528  ‘cfv 6533  Xcixp 8904

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ixp 8905

This theorem is used by:  ixpprc  8926  undifixp  8941  resixpfo  8943  boxcutc  8948  ixpiunwdom  9562  prdsbasfn  17556  xpsff1o  17653  sscfn1  17906  funcfn2  17958  natfn  18046  pthaus  23864  ptuncnv  24033  ptunhmeo  24034  ptcmplem2  24279  prdsbl  24717  finixpnum  38359  upixp  38479  prdstotbnd  38544  elixpconstg  45921  rrxsnicc  47128  ioorrnopnxrlem  47134  hoidmvlelem3  47425  hspdifhsp  47444  hspmbllem2  47455