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Theorem uniixp 8933
Description: The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
uniixp X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem uniixp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ixpf 8932 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
2 fssxp 6745 . . . . 5 (𝑓:𝐴 𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
31, 2syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
4 velpw 4603 . . . 4 (𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
53, 4sylibr 233 . . 3 (𝑓X𝑥𝐴 𝐵𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵))
65ssriv 3982 . 2 X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵)
7 sspwuni 5097 . 2 (X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵))
86, 7mpbi 229 1 X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  wss 3945  𝒫 cpw 4598   cuni 4903   ciun 4991   × cxp 5670  wf 6538  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-pw 4600  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:  ixpexg  8934
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