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Theorem iunxpf 4829
 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, zC = 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
21nfel2 2501 . . . 4 y w C
3 iunxpf.2 . . . . 5 zC
43nfel2 2501 . . . 4 z w C
5 iunxpf.3 . . . . 5 xD
65nfel2 2501 . . . 4 x w D
7 iunxpf.4 . . . . 5 (x = y, zC = D)
87eleq2d 2420 . . . 4 (x = y, z → (w Cw D))
92, 4, 6, 8rexxpf 4828 . . 3 (x (A × B)w Cy A z B w D)
10 eliun 3973 . . 3 (w x (A × B)Cx (A × B)w C)
11 eliun 3973 . . . 4 (w y A z B Dy A w z B D)
12 eliun 3973 . . . . 5 (w z B Dz B w D)
1312rexbii 2639 . . . 4 (y A w z B Dy A z B w D)
1411, 13bitri 240 . . 3 (w y A z B Dy A z B w D)
159, 10, 143bitr4i 268 . 2 (w x (A × B)Cw y A z B D)
1615eqriv 2350 1 x (A × B)C = y A z B D
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  ∃wrex 2615  ∪ciun 3969  ⟨cop 4561   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-xp 4784 This theorem is referenced by: (None)
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