Theorem ixpfn 8879

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 6612	. 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
2		elixp2 8877	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3	2	simp2bi 1146	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴)
4	1, 3	vtoclga 3546	1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   Fn wfn 6509  ‘cfv 6514  Xcixp 8873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-ixp 8874

This theorem is referenced by:  ixpprc  8895  undifixp  8910  resixpfo  8912  boxcutc  8917  ixpiunwdom  9550  prdsbasfn  17441  xpsff1o  17537  sscfn1  17786  funcfn2  17838  natfn  17926  pthaus  23532  ptuncnv  23701  ptunhmeo  23702  ptcmplem2  23947  prdsbl  24386  finixpnum  37606  upixp  37730  prdstotbnd  37795  elixpconstg  45090  rrxsnicc  46305  ioorrnopnxrlem  46311  hoidmvlelem3  46602  hspdifhsp  46621  hspmbllem2  46632