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Theorem xpexgALT 7109
Description: Alternate proof of xpexg 6914 requiring Replacement (ax-rep 4736) but not Power Set (ax-pow 4808). (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 4546 . . . 4 𝑦𝐵 {𝑦} = 𝐵
21xpeq2i 5101 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = (𝐴 × 𝐵)
3 xpiundi 5139 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = 𝑦𝐵 (𝐴 × {𝑦})
42, 3eqtr3i 2650 . 2 (𝐴 × 𝐵) = 𝑦𝐵 (𝐴 × {𝑦})
5 id 22 . . 3 (𝐵𝑊𝐵𝑊)
6 fconstmpt 5128 . . . . 5 (𝐴 × {𝑦}) = (𝑥𝐴𝑦)
7 mptexg 6439 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝑦) ∈ V)
86, 7syl5eqel 2708 . . . 4 (𝐴𝑉 → (𝐴 × {𝑦}) ∈ V)
98ralrimivw 2966 . . 3 (𝐴𝑉 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 iunexg 7092 . . 3 ((𝐵𝑊 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
115, 9, 10syl2anr 495 . 2 ((𝐴𝑉𝐵𝑊) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
124, 11syl5eqel 2708 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1992  wral 2912  Vcvv 3191  {csn 4153   ciun 4490  cmpt 4678   × cxp 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858
This theorem is referenced by: (None)
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