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Theorem ixpf 6776
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 6758 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssiun2 3956 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
32sseld 3179 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵 → (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
43ralimia 2555 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵)
54anim2i 342 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
6 nfcv 2336 . . . . 5 𝑥𝐴
7 nfiu1 3943 . . . . 5 𝑥 𝑥𝐴 𝐵
8 nfcv 2336 . . . . 5 𝑥𝐹
96, 7, 8ffnfvf 5718 . . . 4 (𝐹:𝐴 𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
105, 9sylibr 134 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴 𝑥𝐴 𝐵)
11103adant1 1017 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴 𝑥𝐴 𝐵)
121, 11sylbi 121 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980  wcel 2164  wral 2472  Vcvv 2760   ciun 3913   Fn wfn 5250  wf 5251  cfv 5255  Xcixp 6754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ixp 6755
This theorem is referenced by:  uniixp  6777  ixpssmap2g  6783
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