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Theorem xpexgALT 7492
Description: Alternate proof of xpexg 7288 requiring Replacement (ax-rep 5045) but not Power Set (ax-pow 5115). (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 4846 . . . 4 𝑦𝐵 {𝑦} = 𝐵
21xpeq2i 5430 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = (𝐴 × 𝐵)
3 xpiundi 5468 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = 𝑦𝐵 (𝐴 × {𝑦})
42, 3eqtr3i 2798 . 2 (𝐴 × 𝐵) = 𝑦𝐵 (𝐴 × {𝑦})
5 id 22 . . 3 (𝐵𝑊𝐵𝑊)
6 fconstmpt 5460 . . . . 5 (𝐴 × {𝑦}) = (𝑥𝐴𝑦)
7 mptexg 6808 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝑦) ∈ V)
86, 7syl5eqel 2864 . . . 4 (𝐴𝑉 → (𝐴 × {𝑦}) ∈ V)
98ralrimivw 3127 . . 3 (𝐴𝑉 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 iunexg 7474 . . 3 ((𝐵𝑊 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
115, 9, 10syl2anr 587 . 2 ((𝐴𝑉𝐵𝑊) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
124, 11syl5eqel 2864 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2050  wral 3082  Vcvv 3409  {csn 4435   ciun 4788  cmpt 5004   × cxp 5401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193
This theorem is referenced by: (None)
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