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Theorem iunxpf 4411
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 2154 . . . 4 y w 𝐶
3 iunxpf.2 . . . . 5 z𝐶
43nfcri 2154 . . . 4 z w 𝐶
5 iunxpf.3 . . . . 5 x𝐷
65nfcri 2154 . . . 4 x w 𝐷
7 iunxpf.4 . . . . 5 (x = ⟨y, z⟩ → 𝐶 = 𝐷)
87eleq2d 2089 . . . 4 (x = ⟨y, z⟩ → (w 𝐶w 𝐷))
92, 4, 6, 8rexxpf 4410 . . 3 (x (A × B)w 𝐶y A z B w 𝐷)
10 eliun 3635 . . 3 (w x (A × B)𝐶x (A × B)w 𝐶)
11 eliun 3635 . . . 4 (w y A z B 𝐷y A w z B 𝐷)
12 eliun 3635 . . . . 5 (w z B 𝐷z B w 𝐷)
1312rexbii 2309 . . . 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 2019 1 x (A × B)𝐶 = y A z B 𝐷
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  wnfc 2147  wrex 2285  cop 3353   ciun 3631   × cxp 4270
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-opab 3793  df-xp 4278  df-rel 4279
This theorem is referenced by:  dfmpt2  5767
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