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Theorem uniixp 8684
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 8683 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
2 fssxp 6625 . . . . 5 (𝑓:𝐴 𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
31, 2syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
4 velpw 4544 . . . 4 (𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
53, 4sylibr 233 . . 3 (𝑓X𝑥𝐴 𝐵𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵))
65ssriv 3930 . 2 X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵)
7 sspwuni 5034 . 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 2110  wss 3892  𝒫 cpw 4539   cuni 4845   ciun 4930   × cxp 5587  wf 6427  Xcixp 8660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-fv 6439  df-ixp 8661
This theorem is referenced by:  ixpexg  8685
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