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Theorem ixpintm 6715
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 6702 . . 3 (∀𝑥𝐴 𝐵 = 𝑦𝐵 𝑦X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦)
2 intiin 3936 . . . 4 𝐵 = 𝑦𝐵 𝑦
32a1i 9 . . 3 (𝑥𝐴 𝐵 = 𝑦𝐵 𝑦)
41, 3mprg 2532 . 2 X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦
5 ixpiinm 6714 . 2 (∃𝑧 𝑧𝐵X𝑥𝐴 𝑦𝐵 𝑦 = 𝑦𝐵 X𝑥𝐴 𝑦)
64, 5eqtrid 2220 1 (∃𝑧 𝑧𝐵X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wex 1490  wcel 2146   cint 3840   ciin 3883  Xcixp 6688
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iin 3885  df-br 3999  df-opab 4060  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ixp 6689
This theorem is referenced by: (None)
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