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Theorem ixpf 8932
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 8913 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssiun2 5044 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
32sseld 3977 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵 → (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
43ralimia 3076 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵)
54anim2i 616 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
6 nfcv 2899 . . . . 5 𝑥𝐴
7 nfiu1 5025 . . . . 5 𝑥 𝑥𝐴 𝐵
8 nfcv 2899 . . . . 5 𝑥𝐹
96, 7, 8ffnfvf 7124 . . . 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 1128 . 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 395  w3a 1085  wcel 2099  wral 3057  Vcvv 3470   ciun 4991   Fn wfn 6537  wf 6538  cfv 6542  Xcixp 8909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ixp 8910
This theorem is referenced by:  uniixp  8933  ixpssmap2g  8939  ioorrnopnlem  45686  iunhoiioolem  46057
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