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Theorem ixpf 8911
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 8892 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssiun2 5041 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
32sseld 3974 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵 → (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
43ralimia 3072 . . . . 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 2895 . . . . 5 𝑥𝐴
7 nfiu1 5022 . . . . 5 𝑥 𝑥𝐴 𝐵
8 nfcv 2895 . . . . 5 𝑥𝐹
96, 7, 8ffnfvf 7112 . . . 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 1127 . 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 1084  wcel 2098  wral 3053  Vcvv 3466   ciun 4988   Fn wfn 6529  wf 6530  cfv 6534  Xcixp 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ixp 8889
This theorem is referenced by:  uniixp  8912  ixpssmap2g  8918  ioorrnopnlem  45530  iunhoiioolem  45901
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