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Theorem coiun 4773
 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 4762 . 2 Rel (A x 𝐶 B)
2 reliun 4401 . . 3 (Rel x 𝐶 (AB) ↔ x 𝐶 Rel (AB))
3 relco 4762 . . . 4 Rel (AB)
43a1i 9 . . 3 (x 𝐶 → Rel (AB))
52, 4mprgbir 2373 . 2 Rel x 𝐶 (AB)
6 eliun 3652 . . . . . . . 8 (⟨y, w x 𝐶 Bx 𝐶y, w B)
7 df-br 3756 . . . . . . . 8 (y x 𝐶 Bw ↔ ⟨y, w x 𝐶 B)
8 df-br 3756 . . . . . . . . 9 (yBw ↔ ⟨y, w B)
98rexbii 2325 . . . . . . . 8 (x 𝐶 yBwx 𝐶y, w B)
106, 7, 93bitr4i 201 . . . . . . 7 (y x 𝐶 Bwx 𝐶 yBw)
1110anbi1i 431 . . . . . 6 ((y x 𝐶 Bw wAz) ↔ (x 𝐶 yBw wAz))
12 r19.41v 2460 . . . . . 6 (x 𝐶 (yBw wAz) ↔ (x 𝐶 yBw wAz))
1311, 12bitr4i 176 . . . . 5 ((y x 𝐶 Bw wAz) ↔ x 𝐶 (yBw wAz))
1413exbii 1493 . . . 4 (w(y x 𝐶 Bw wAz) ↔ wx 𝐶 (yBw wAz))
15 rexcom4 2571 . . . 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 2554 . . . 4 y V
18 vex 2554 . . . 4 z V
1917, 18opelco 4450 . . 3 (⟨y, z (A x 𝐶 B) ↔ w(y x 𝐶 Bw wAz))
20 eliun 3652 . . . 4 (⟨y, z x 𝐶 (AB) ↔ x 𝐶y, z (AB))
2117, 18opelco 4450 . . . . 5 (⟨y, z (AB) ↔ w(yBw wAz))
2221rexbii 2325 . . . 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 4377 1 (A x 𝐶 B) = x 𝐶 (AB)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370  ∪ ciun 3648   class class class wbr 3755   ∘ ccom 4292  Rel wrel 4293 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-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  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-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-co 4297 This theorem is referenced by: (None)
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