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Theorem xpexgALT 5791
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 4480 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 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)

Proof of Theorem xpexgALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunid 3741 . . . 4 𝑦𝐵 {𝑦} = 𝐵
21xpeq2i 4392 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = (𝐴 × 𝐵)
3 xpiundi 4424 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = 𝑦𝐵 (𝐴 × {𝑦})
42, 3eqtr3i 2104 . 2 (𝐴 × 𝐵) = 𝑦𝐵 (𝐴 × {𝑦})
5 id 19 . . 3 (𝐵𝑊𝐵𝑊)
6 fconstmpt 4413 . . . . 5 (𝐴 × {𝑦}) = (𝑥𝐴𝑦)
7 mptexg 5418 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝑦) ∈ V)
86, 7syl5eqel 2166 . . . 4 (𝐴𝑉 → (𝐴 × {𝑦}) ∈ V)
98ralrimivw 2436 . . 3 (𝐴𝑉 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 iunexg 5777 . . 3 ((𝐵𝑊 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
115, 9, 10syl2anr 284 . 2 ((𝐴𝑉𝐵𝑊) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
124, 11syl5eqel 2166 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  wral 2349  Vcvv 2602  {csn 3406   ciun 3686  cmpt 3847   × cxp 4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940
This theorem is referenced by: (None)
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