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Theorem iunxpf 4427
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 = y A z B 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 2169 . . . 4 y w 𝐶
3 iunxpf.2 . . . . 5 z𝐶
43nfcri 2169 . . . 4 z w 𝐶
5 iunxpf.3 . . . . 5 x𝐷
65nfcri 2169 . . . 4 x w 𝐷
7 iunxpf.4 . . . . 5 (x = ⟨y, z⟩ → 𝐶 = 𝐷)
87eleq2d 2104 . . . 4 (x = ⟨y, z⟩ → (w 𝐶w 𝐷))
92, 4, 6, 8rexxpf 4426 . . 3 (x (A × B)w 𝐶y A z B w 𝐷)
10 eliun 3652 . . 3 (w x (A × B)𝐶x (A × B)w 𝐶)
11 eliun 3652 . . . 4 (w y A z B 𝐷y A w z B 𝐷)
12 eliun 3652 . . . . 5 (w z B 𝐷z B w 𝐷)
1312rexbii 2325 . . . 4 (y A w z B 𝐷y A z B w 𝐷)
1411, 13bitri 173 . . 3 (w y A z B 𝐷y A z B w 𝐷)
159, 10, 143bitr4i 201 . 2 (w x (A × B)𝐶w y A z B 𝐷)
1615eqriv 2034 1 x (A × B)𝐶 = y A z B 𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wnfc 2162  wrex 2301  cop 3370   ciun 3648   × cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  dfmpt2  5786
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