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Theorem xpexgALT 5702
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4395 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpexgALT ((A 𝑉 B 𝑊) → (A × B) V)

Proof of Theorem xpexgALT
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunid 3703 . . . 4 y B {y} = B
21xpeq2i 4309 . . 3 (A × y B {y}) = (A × B)
3 xpiundi 4341 . . 3 (A × y B {y}) = y B (A × {y})
42, 3eqtr3i 2059 . 2 (A × B) = y B (A × {y})
5 id 19 . . 3 (B 𝑊B 𝑊)
6 fconstmpt 4330 . . . . 5 (A × {y}) = (x Ay)
7 mptexg 5329 . . . . 5 (A 𝑉 → (x Ay) V)
86, 7syl5eqel 2121 . . . 4 (A 𝑉 → (A × {y}) V)
98ralrimivw 2387 . . 3 (A 𝑉y B (A × {y}) V)
10 iunexg 5688 . . 3 ((B 𝑊 y B (A × {y}) V) → y B (A × {y}) V)
115, 9, 10syl2anr 274 . 2 ((A 𝑉 B 𝑊) → y B (A × {y}) V)
124, 11syl5eqel 2121 1 ((A 𝑉 B 𝑊) → (A × B) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  Vcvv 2551  {csn 3367   ciun 3648  cmpt 3809   × 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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
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-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by: (None)
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