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Theorem uniixp 6714
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 6713 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
2 fssxp 5378 . . . . 5 (𝑓:𝐴 𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
31, 2syl 14 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
4 velpw 3581 . . . 4 (𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
53, 4sylibr 134 . . 3 (𝑓X𝑥𝐴 𝐵𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵))
65ssriv 3159 . 2 X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵)
7 sspwuni 3968 . 2 (X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵))
86, 7mpbi 145 1 X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wcel 2148  wss 3129  𝒫 cpw 3574   cuni 3807   ciun 3884   × cxp 4620  wf 5207  Xcixp 6691
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ixp 6692
This theorem is referenced by:  ixpexgg  6715
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