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Theorem iunxpf 5712
Description: Indexed union on a Cartesian product 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 2968 . . . 4 𝑦 𝑤𝐶
3 iunxpf.2 . . . . 5 𝑧𝐶
43nfcri 2968 . . . 4 𝑧 𝑤𝐶
5 iunxpf.3 . . . . 5 𝑥𝐷
65nfcri 2968 . . . 4 𝑥 𝑤𝐷
7 iunxpf.4 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
87eleq2d 2895 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤𝐶𝑤𝐷))
92, 4, 6, 8rexxpf 5711 . . 3 (∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
10 eliun 4914 . . 3 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶)
11 eliun 4914 . . . 4 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴 𝑤 𝑧𝐵 𝐷)
12 eliun 4914 . . . . 5 (𝑤 𝑧𝐵 𝐷 ↔ ∃𝑧𝐵 𝑤𝐷)
1312rexbii 3244 . . . 4 (∃𝑦𝐴 𝑤 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
1411, 13bitri 276 . . 3 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
159, 10, 143bitr4i 304 . 2 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶𝑤 𝑦𝐴 𝑧𝐵 𝐷)
1615eqriv 2815 1 𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wnfc 2958  wrex 3136  cop 4563   ciun 4910   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4912  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  dfmpo  7786
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