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Theorem iunxpf 4757
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 2306 . . . 4 𝑦 𝑤𝐶
3 iunxpf.2 . . . . 5 𝑧𝐶
43nfcri 2306 . . . 4 𝑧 𝑤𝐶
5 iunxpf.3 . . . . 5 𝑥𝐷
65nfcri 2306 . . . 4 𝑥 𝑤𝐷
7 iunxpf.4 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
87eleq2d 2240 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤𝐶𝑤𝐷))
92, 4, 6, 8rexxpf 4756 . . 3 (∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶 ↔ ∃𝑦𝐴𝑧𝐵 𝑤𝐷)
10 eliun 3875 . . 3 (𝑤 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤𝐶)
11 eliun 3875 . . . 4 (𝑤 𝑦𝐴 𝑧𝐵 𝐷 ↔ ∃𝑦𝐴 𝑤 𝑧𝐵 𝐷)
12 eliun 3875 . . . . 5 (𝑤 𝑧𝐵 𝐷 ↔ ∃𝑧𝐵 𝑤𝐷)
1312rexbii 2477 . . . 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 2167 1 𝑥 ∈ (𝐴 × 𝐵)𝐶 = 𝑦𝐴 𝑧𝐵 𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  wnfc 2299  wrex 2449  cop 3584   ciun 3871   × cxp 4607
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-iun 3873  df-opab 4049  df-xp 4615  df-rel 4616
This theorem is referenced by:  dfmpo  6199
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