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Theorem iunxpf 4694
Description: Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
iunxpf.1 𝑦𝐶
iunxpf.2 𝑧𝐶
iunxpf.3 𝑥𝐷
iunxpf.4 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
iunxpf 𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝐵,𝑦
Allowed substitution hints:   𝐴(𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥,𝑦,𝑧)

Proof of Theorem iunxpf
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 iunxpf.1 . . . . 5 𝑦𝐶
21nfcri 2276 . . . 4 𝑦 𝑤𝐶
3 iunxpf.2 . . . . 5 𝑧𝐶
43nfcri 2276 . . . 4 𝑧 𝑤𝐶
5 iunxpf.3 . . . . 5 𝑥𝐷
65nfcri 2276 . . . 4 𝑥 𝑤𝐷
7 iunxpf.4 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
87eleq2d 2210 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤𝐶𝑤𝐷))
92, 4, 6, 8rexxpf 4693 . . 3 (∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
10 eliun 3824 . . 3 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶)
11 eliun 3824 . . . 4 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴 𝑤 𝑧𝐵 𝐷)
12 eliun 3824 . . . . 5 (𝑤 𝑧𝐵 𝐷 ↔ ∃𝑧𝐵 𝑤𝐷)
1312rexbii 2445 . . . 4 (∃𝑦𝐴 𝑤 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
1411, 13bitri 183 . . 3 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
159, 10, 143bitr4i 211 . 2 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶𝑤 𝑦𝐴 𝑧𝐵 𝐷)
1615eqriv 2137 1 𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  wcel 1481  wnfc 2269  wrex 2418  cop 3534   ciun 3820   × cxp 4544
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-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-iun 3822  df-opab 3997  df-xp 4552  df-rel 4553
This theorem is referenced by:  dfmpo  6127
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