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Theorem uniixp 8085
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 8084 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
2 fssxp 6209 . . . . 5 (𝑓:𝐴 𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
31, 2syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
4 selpw 4297 . . . 4 (𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
53, 4sylibr 224 . . 3 (𝑓X𝑥𝐴 𝐵𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵))
65ssriv 3736 . 2 X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵)
7 sspwuni 4751 . 2 (X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵))
86, 7mpbi 220 1 X𝑥𝐴 𝐵 ⊆ (𝐴 × 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2127  wss 3703  𝒫 cpw 4290   cuni 4576   ciun 4660   × cxp 5252  wf 6033  Xcixp 8062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ixp 8063
This theorem is referenced by:  ixpexg  8086
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