MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpexgALT Structured version   Visualization version   GIF version

Theorem xpexgALT 7915
Description: Alternate proof of xpexg 7685 requiring Replacement (ax-rep 5243) but not Power Set (ax-pow 5321). (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 5021 . . . 4 𝑦𝐵 {𝑦} = 𝐵
21xpeq2i 5661 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = (𝐴 × 𝐵)
3 xpiundi 5703 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = 𝑦𝐵 (𝐴 × {𝑦})
42, 3eqtr3i 2767 . 2 (𝐴 × 𝐵) = 𝑦𝐵 (𝐴 × {𝑦})
5 id 22 . . 3 (𝐵𝑊𝐵𝑊)
6 fconstmpt 5695 . . . . 5 (𝐴 × {𝑦}) = (𝑥𝐴𝑦)
7 mptexg 7172 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝑦) ∈ V)
86, 7eqeltrid 2842 . . . 4 (𝐴𝑉 → (𝐴 × {𝑦}) ∈ V)
98ralrimivw 3148 . . 3 (𝐴𝑉 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 iunexg 7897 . . 3 ((𝐵𝑊 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
115, 9, 10syl2anr 598 . 2 ((𝐴𝑉𝐵𝑊) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
124, 11eqeltrid 2842 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3065  Vcvv 3446  {csn 4587   ciun 4955  cmpt 5189   × cxp 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator