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Theorem ixpf 8861
Description: A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
Assertion
Ref Expression
ixpf (𝐹X𝑥𝐴 𝐵𝐹:𝐴 𝑥𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ixpf
StepHypRef Expression
1 elixp2 8842 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssiun2 5008 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
32sseld 3944 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵 → (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
43ralimia 3080 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵)
54anim2i 618 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
6 nfcv 2904 . . . . 5 𝑥𝐴
7 nfiu1 4989 . . . . 5 𝑥 𝑥𝐴 𝐵
8 nfcv 2904 . . . . 5 𝑥𝐹
96, 7, 8ffnfvf 7068 . . . 4 (𝐹:𝐴 𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
105, 9sylibr 233 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴 𝑥𝐴 𝐵)
11103adant1 1131 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴 𝑥𝐴 𝐵)
121, 11sylbi 216 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wral 3061  Vcvv 3444   ciun 4955   Fn wfn 6492  wf 6493  cfv 6497  Xcixp 8838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ixp 8839
This theorem is referenced by:  uniixp  8862  ixpssmap2g  8868  ioorrnopnlem  44631  iunhoiioolem  45002
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