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Theorem iunxpf 5805
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 2891 . . . 4 𝑦 𝑤𝐶
3 iunxpf.2 . . . . 5 𝑧𝐶
43nfcri 2891 . . . 4 𝑧 𝑤𝐶
5 iunxpf.3 . . . . 5 𝑥𝐷
65nfcri 2891 . . . 4 𝑥 𝑤𝐷
7 iunxpf.4 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
87eleq2d 2820 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤𝐶𝑤𝐷))
92, 4, 6, 8rexxpf 5804 . . 3 (∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
10 eliun 4959 . . 3 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶)
11 eliun 4959 . . . 4 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴 𝑤 𝑧𝐵 𝐷)
12 eliun 4959 . . . . 5 (𝑤 𝑧𝐵 𝐷 ↔ ∃𝑧𝐵 𝑤𝐷)
1312rexbii 3094 . . . 4 (∃𝑦𝐴 𝑤 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
1411, 13bitri 275 . . 3 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
159, 10, 143bitr4i 303 . 2 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶𝑤 𝑦𝐴 𝑧𝐵 𝐷)
1615eqriv 2730 1 𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wnfc 2884  wrex 3070  cop 4593   ciun 4955   × cxp 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4957  df-opab 5169  df-xp 5640  df-rel 5641
This theorem is referenced by:  dfmpo  8035
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