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Theorem xpiundir 4342
Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
Assertion
Ref Expression
xpiundir ( x A B × 𝐶) = x A (B × 𝐶)
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)

Proof of Theorem xpiundir
Dummy variables y w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2571 . . . . 5 (x A y(y B w 𝐶 z = ⟨y, w⟩) ↔ yx A (y B w 𝐶 z = ⟨y, w⟩))
2 df-rex 2306 . . . . . 6 (y B w 𝐶 z = ⟨y, w⟩ ↔ y(y B w 𝐶 z = ⟨y, w⟩))
32rexbii 2325 . . . . 5 (x A y B w 𝐶 z = ⟨y, w⟩ ↔ x A y(y B w 𝐶 z = ⟨y, w⟩))
4 eliun 3652 . . . . . . . 8 (y x A Bx A y B)
54anbi1i 431 . . . . . . 7 ((y x A B w 𝐶 z = ⟨y, w⟩) ↔ (x A y B w 𝐶 z = ⟨y, w⟩))
6 r19.41v 2460 . . . . . . 7 (x A (y B w 𝐶 z = ⟨y, w⟩) ↔ (x A y B w 𝐶 z = ⟨y, w⟩))
75, 6bitr4i 176 . . . . . 6 ((y x A B w 𝐶 z = ⟨y, w⟩) ↔ x A (y B w 𝐶 z = ⟨y, w⟩))
87exbii 1493 . . . . 5 (y(y x A B w 𝐶 z = ⟨y, w⟩) ↔ yx A (y B w 𝐶 z = ⟨y, w⟩))
91, 3, 83bitr4ri 202 . . . 4 (y(y x A B w 𝐶 z = ⟨y, w⟩) ↔ x A y B w 𝐶 z = ⟨y, w⟩)
10 df-rex 2306 . . . 4 (y x A Bw 𝐶 z = ⟨y, w⟩ ↔ y(y x A B w 𝐶 z = ⟨y, w⟩))
11 elxp2 4306 . . . . 5 (z (B × 𝐶) ↔ y B w 𝐶 z = ⟨y, w⟩)
1211rexbii 2325 . . . 4 (x A z (B × 𝐶) ↔ x A y B w 𝐶 z = ⟨y, w⟩)
139, 10, 123bitr4i 201 . . 3 (y x A Bw 𝐶 z = ⟨y, w⟩ ↔ x A z (B × 𝐶))
14 elxp2 4306 . . 3 (z ( x A B × 𝐶) ↔ y x A Bw 𝐶 z = ⟨y, w⟩)
15 eliun 3652 . . 3 (z x A (B × 𝐶) ↔ x A z (B × 𝐶))
1613, 14, 153bitr4i 201 . 2 (z ( x A B × 𝐶) ↔ z x A (B × 𝐶))
1716eqriv 2034 1 ( x A B × 𝐶) = x A (B × 𝐶)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370   ciun 3648   × cxp 4286
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-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-opab 3810  df-xp 4294
This theorem is referenced by:  iunxpconst  4343  resiun2  4574
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