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Theorem ixpintm 6585
 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 6572 . . 3 (∀𝑥𝐴 𝐵 = 𝑦𝐵 𝑦X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦)
2 intiin 3835 . . . 4 𝐵 = 𝑦𝐵 𝑦
32a1i 9 . . 3 (𝑥𝐴 𝐵 = 𝑦𝐵 𝑦)
41, 3mprg 2464 . 2 X𝑥𝐴 𝐵 = X𝑥𝐴 𝑦𝐵 𝑦
5 ixpiinm 6584 . 2 (∃𝑧 𝑧𝐵X𝑥𝐴 𝑦𝐵 𝑦 = 𝑦𝐵 X𝑥𝐴 𝑦)
64, 5syl5eq 2160 1 (∃𝑧 𝑧𝐵X𝑥𝐴 𝐵 = 𝑦𝐵 X𝑥𝐴 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1314  ∃wex 1451   ∈ wcel 1463  ∩ cint 3739  ∩ ciin 3782  Xcixp 6558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iin 3784  df-br 3898  df-opab 3958  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099  df-ixp 6559 This theorem is referenced by: (None)
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