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Theorem xpiundir 4326
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 2554 . . . . 5 (x A y(y B w 𝐶 z = ⟨y, w⟩) ↔ yx A (y B w 𝐶 z = ⟨y, w⟩))
2 df-rex 2290 . . . . . 6 (y B w 𝐶 z = ⟨y, w⟩ ↔ y(y B w 𝐶 z = ⟨y, w⟩))
32rexbii 2309 . . . . 5 (x A y B w 𝐶 z = ⟨y, w⟩ ↔ x A y(y B w 𝐶 z = ⟨y, w⟩))
4 eliun 3635 . . . . . . . 8 (y x A Bx A y B)
54anbi1i 434 . . . . . . 7 ((y x A B w 𝐶 z = ⟨y, w⟩) ↔ (x A y B w 𝐶 z = ⟨y, w⟩))
6 r19.41v 2444 . . . . . . 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 1478 . . . . 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 2290 . . . 4 (y x A Bw 𝐶 z = ⟨y, w⟩ ↔ y(y x A B w 𝐶 z = ⟨y, w⟩))
11 elxp2 4290 . . . . 5 (z (B × 𝐶) ↔ y B w 𝐶 z = ⟨y, w⟩)
1211rexbii 2309 . . . 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 4290 . . 3 (z ( x A B × 𝐶) ↔ y x A Bw 𝐶 z = ⟨y, w⟩)
15 eliun 3635 . . 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 2019 1 ( x A B × 𝐶) = x A (B × 𝐶)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  wrex 2285  cop 3353   ciun 3631   × cxp 4270
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-opab 3793  df-xp 4278
This theorem is referenced by:  iunxpconst  4327  resiun2  4558
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