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Theorem iunxpf 5687
 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 2946 . . . 4 𝑦 𝑤𝐶
3 iunxpf.2 . . . . 5 𝑧𝐶
43nfcri 2946 . . . 4 𝑧 𝑤𝐶
5 iunxpf.3 . . . . 5 𝑥𝐷
65nfcri 2946 . . . 4 𝑥 𝑤𝐷
7 iunxpf.4 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
87eleq2d 2878 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤𝐶𝑤𝐷))
92, 4, 6, 8rexxpf 5686 . . 3 (∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
10 eliun 4888 . . 3 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶)
11 eliun 4888 . . . 4 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴 𝑤 𝑧𝐵 𝐷)
12 eliun 4888 . . . . 5 (𝑤 𝑧𝐵 𝐷 ↔ ∃𝑧𝐵 𝑤𝐷)
1312rexbii 3213 . . . 4 (∃𝑦𝐴 𝑤 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
1411, 13bitri 278 . . 3 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
159, 10, 143bitr4i 306 . 2 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶𝑤 𝑦𝐴 𝑧𝐵 𝐷)
1615eqriv 2798 1 𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  Ⅎwnfc 2939  ∃wrex 3110  ⟨cop 4534  ∪ ciun 4884   × cxp 5521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-iun 4886  df-opab 5096  df-xp 5529  df-rel 5530 This theorem is referenced by:  dfmpo  7784
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