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Theorem ixpintm 6739
Description: The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
Assertion
Ref Expression
ixpintm (∃𝑧 𝑧𝐵X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝑧,𝐵
Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem ixpintm
StepHypRef Expression
1 ixpeq2 6726 . . 3 (∀𝑥𝐴 𝐵 = 𝑦𝐵 𝑦X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦)
2 intiin 3953 . . . 4 𝐵 = 𝑦𝐵 𝑦
32a1i 9 . . 3 (𝑥𝐴 𝐵 = 𝑦𝐵 𝑦)
41, 3mprg 2544 . 2 X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦
5 ixpiinm 6738 . 2 (∃𝑧 𝑧𝐵X𝑥𝐴 𝑦𝐵 𝑦 = 𝑦𝐵 X𝑥𝐴 𝑦)
64, 5eqtrid 2232 1 (∃𝑧 𝑧𝐵X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wex 1502  wcel 2158   cint 3856   ciin 3899  Xcixp 6712
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iin 3901  df-br 4016  df-opab 4077  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ixp 6713
This theorem is referenced by: (None)
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