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Theorem ixpiinm 6622
Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
Assertion
Ref Expression
ixpiinm (∃𝑧 𝑧𝐵X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝑧,𝐵
Allowed substitution hints:   𝐴(𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem ixpiinm
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2201 . . . 4 (𝑦 = 𝑧 → (𝑦𝐵𝑧𝐵))
21cbvexv 1891 . . 3 (∃𝑦 𝑦𝐵 ↔ ∃𝑧 𝑧𝐵)
3 r19.28mv 3456 . . . . 5 (∃𝑦 𝑦𝐵 → (∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
4 eliin 3822 . . . . . . 7 (𝑓 ∈ V → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶))
54elv 2691 . . . . . 6 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑓X𝑥𝐴 𝐶)
6 vex 2690 . . . . . . . 8 𝑓 ∈ V
76elixp 6603 . . . . . . 7 (𝑓X𝑥𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
87ralbii 2442 . . . . . 6 (∀𝑦𝐵 𝑓X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
95, 8bitri 183 . . . . 5 (𝑓 𝑦𝐵 X𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
106elixp 6603 . . . . . 6 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶))
11 vex 2690 . . . . . . . . . . 11 𝑥 ∈ V
126, 11fvex 5445 . . . . . . . . . 10 (𝑓𝑥) ∈ V
13 eliin 3822 . . . . . . . . . 10 ((𝑓𝑥) ∈ V → ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶))
1412, 13ax-mp 5 . . . . . . . . 9 ((𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
1514ralbii 2442 . . . . . . . 8 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶)
16 ralcom 2595 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1715, 16bitri 183 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶)
1817anbi2i 453 . . . . . 6 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑦𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
1910, 18bitri 183 . . . . 5 (𝑓X𝑥𝐴 𝑦𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐵𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
203, 9, 193bitr4g 222 . . . 4 (∃𝑦 𝑦𝐵 → (𝑓 𝑦𝐵 X𝑥𝐴 𝐶𝑓X𝑥𝐴 𝑦𝐵 𝐶))
2120eqrdv 2138 . . 3 (∃𝑦 𝑦𝐵 𝑦𝐵 X𝑥𝐴 𝐶 = X𝑥𝐴 𝑦𝐵 𝐶)
222, 21sylbir 134 . 2 (∃𝑧 𝑧𝐵 𝑦𝐵 X𝑥𝐴 𝐶 = X𝑥𝐴 𝑦𝐵 𝐶)
2322eqcomd 2146 1 (∃𝑧 𝑧𝐵X𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 X𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481  wral 2417  Vcvv 2687   ciin 3818   Fn wfn 5122  cfv 5127  Xcixp 6596
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iin 3820  df-br 3934  df-opab 3994  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-iota 5092  df-fun 5129  df-fn 5130  df-fv 5135  df-ixp 6597
This theorem is referenced by:  ixpintm  6623
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