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Theorem uniixp 7875
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 7874 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
2 fssxp 6017 . . . . 5 (𝑓:𝐴 𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
31, 2syl 17 . . . 4 (𝑓X𝑥𝐴 𝐵𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
4 selpw 4137 . . . 4 (𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵) ↔ 𝑓 ⊆ (𝐴 × 𝑥𝐴 𝐵))
53, 4sylibr 224 . . 3 (𝑓X𝑥𝐴 𝐵𝑓 ∈ 𝒫 (𝐴 × 𝑥𝐴 𝐵))
65ssriv 3587 . 2 X𝑥𝐴 𝐵 ⊆ 𝒫 (𝐴 × 𝑥𝐴 𝐵)
7 sspwuni 4577 . 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 1987  wss 3555  𝒫 cpw 4130   cuni 4402   ciun 4485   × cxp 5072  wf 5843  Xcixp 7852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ixp 7853
This theorem is referenced by:  ixpexg  7876
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