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Theorem ixpf 6582
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 6564 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 ssiun2 3826 . . . . . . 7 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
32sseld 3066 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) ∈ 𝐵 → (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
43ralimia 2470 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵)
54anim2i 339 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
6 nfcv 2258 . . . . 5 𝑥𝐴
7 nfiu1 3813 . . . . 5 𝑥 𝑥𝐴 𝐵
8 nfcv 2258 . . . . 5 𝑥𝐹
96, 7, 8ffnfvf 5547 . . . 4 (𝐹:𝐴 𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥𝐴 𝐵))
105, 9sylibr 133 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴 𝑥𝐴 𝐵)
11103adant1 984 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴 𝑥𝐴 𝐵)
121, 11sylbi 120 1 (𝐹X𝑥𝐴 𝐵𝐹:𝐴 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947  wcel 1465  wral 2393  Vcvv 2660   ciun 3783   Fn wfn 5088  wf 5089  cfv 5093  Xcixp 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ixp 6561
This theorem is referenced by:  uniixp  6583  ixpssmap2g  6589
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