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Theorem coiun 5090
 Description: Composition with an indexed union. (Contributed by set.mm contributors, 21-Dec-2008.)
Assertion
Ref Expression
coiun (A x C B) = x C (A B)
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   C(x)

Proof of Theorem coiun
Dummy variables w y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 3973 . . . . . . . 8 (y, w x C Bx C y, w B)
2 df-br 4640 . . . . . . . 8 (yx C Bwy, w x C B)
3 df-br 4640 . . . . . . . . 9 (yBwy, w B)
43rexbii 2639 . . . . . . . 8 (x C yBwx C y, w B)
51, 2, 43bitr4i 268 . . . . . . 7 (yx C Bwx C yBw)
65anbi1i 676 . . . . . 6 ((yx C Bw wAz) ↔ (x C yBw wAz))
7 r19.41v 2764 . . . . . 6 (x C (yBw wAz) ↔ (x C yBw wAz))
86, 7bitr4i 243 . . . . 5 ((yx C Bw wAz) ↔ x C (yBw wAz))
98exbii 1582 . . . 4 (w(yx C Bw wAz) ↔ wx C (yBw wAz))
10 rexcom4 2878 . . . 4 (x C w(yBw wAz) ↔ wx C (yBw wAz))
119, 10bitr4i 243 . . 3 (w(yx C Bw wAz) ↔ x C w(yBw wAz))
12 opelco 4884 . . 3 (y, z (A x C B) ↔ w(yx C Bw wAz))
13 eliun 3973 . . . 4 (y, z x C (A B) ↔ x C y, z (A B))
14 opelco 4884 . . . . 5 (y, z (A B) ↔ w(yBw wAz))
1514rexbii 2639 . . . 4 (x C y, z (A B) ↔ x C w(yBw wAz))
1613, 15bitri 240 . . 3 (y, z x C (A B) ↔ x C w(yBw wAz))
1711, 12, 163bitr4i 268 . 2 (y, z (A x C B) ↔ y, z x C (A B))
1817eqrelriv 4850 1 (A x C B) = x C (A B)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∪ciun 3969  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721 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-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-br 4640  df-co 4726 This theorem is referenced by: (None)
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