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Theorem coiun 4753
Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
Assertion
Ref Expression
coiun (A x 𝐶 B) = x 𝐶 (AB)
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 relco 4742 . 2 Rel (A x 𝐶 B)
2 reliun 4381 . . 3 (Rel x 𝐶 (AB) ↔ x 𝐶 Rel (AB))
3 relco 4742 . . . 4 Rel (AB)
43a1i 9 . . 3 (x 𝐶 → Rel (AB))
52, 4mprgbir 2353 . 2 Rel x 𝐶 (AB)
6 eliun 3631 . . . . . . . 8 (⟨y, w x 𝐶 Bx 𝐶y, w B)
7 df-br 3735 . . . . . . . 8 (y x 𝐶 Bw ↔ ⟨y, w x 𝐶 B)
8 df-br 3735 . . . . . . . . 9 (yBw ↔ ⟨y, w B)
98rexbii 2305 . . . . . . . 8 (x 𝐶 yBwx 𝐶y, w B)
106, 7, 93bitr4i 201 . . . . . . 7 (y x 𝐶 Bwx 𝐶 yBw)
1110anbi1i 434 . . . . . 6 ((y x 𝐶 Bw wAz) ↔ (x 𝐶 yBw wAz))
12 r19.41v 2440 . . . . . 6 (x 𝐶 (yBw wAz) ↔ (x 𝐶 yBw wAz))
1311, 12bitr4i 176 . . . . 5 ((y x 𝐶 Bw wAz) ↔ x 𝐶 (yBw wAz))
1413exbii 1474 . . . 4 (w(y x 𝐶 Bw wAz) ↔ wx 𝐶 (yBw wAz))
15 rexcom4 2550 . . . 4 (x 𝐶 w(yBw wAz) ↔ wx 𝐶 (yBw wAz))
1614, 15bitr4i 176 . . 3 (w(y x 𝐶 Bw wAz) ↔ x 𝐶 w(yBw wAz))
17 vex 2534 . . . 4 y V
18 vex 2534 . . . 4 z V
1917, 18opelco 4430 . . 3 (⟨y, z (A x 𝐶 B) ↔ w(y x 𝐶 Bw wAz))
20 eliun 3631 . . . 4 (⟨y, z x 𝐶 (AB) ↔ x 𝐶y, z (AB))
2117, 18opelco 4430 . . . . 5 (⟨y, z (AB) ↔ w(yBw wAz))
2221rexbii 2305 . . . 4 (x 𝐶y, z (AB) ↔ x 𝐶 w(yBw wAz))
2320, 22bitri 173 . . 3 (⟨y, z x 𝐶 (AB) ↔ x 𝐶 w(yBw wAz))
2416, 19, 233bitr4i 201 . 2 (⟨y, z (A x 𝐶 B) ↔ ⟨y, z x 𝐶 (AB))
251, 5, 24eqrelriiv 4357 1 (A x 𝐶 B) = x 𝐶 (AB)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226  wex 1358   wcel 1370  wrex 2281  cop 3349   ciun 3627   class class class wbr 3734  ccom 4272  Rel wrel 4273
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-iun 3629  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-co 4277
This theorem is referenced by: (None)
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