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Theorem iuniin 3694
Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iuniin 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuniin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 r19.12 2439 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 → ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
2 vex 2577 . . . . . 6 𝑧 ∈ V
3 eliin 3689 . . . . . 6 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶))
42, 3ax-mp 7 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶)
54rexbii 2348 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
6 eliun 3688 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
76ralbii 2347 . . . 4 (∀𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
81, 5, 73imtr4i 194 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 → ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
9 eliun 3688 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
10 eliin 3689 . . . 4 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶))
112, 10ax-mp 7 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
128, 9, 113imtr4i 194 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
1312ssriv 2976 1 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
Colors of variables: wff set class
Syntax hints:  wb 102  wcel 1409  wral 2323  wrex 2324  Vcvv 2574  wss 2944   ciun 3684   ciin 3685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958  df-iun 3686  df-iin 3687
This theorem is referenced by: (None)
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