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

Theorem xpexgALT 7677
 Description: Alternate proof of xpexg 7467 requiring Replacement (ax-rep 5176) but not Power Set (ax-pow 5253). (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 4970 . . . 4 𝑦𝐵 {𝑦} = 𝐵
21xpeq2i 5569 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = (𝐴 × 𝐵)
3 xpiundi 5609 . . 3 (𝐴 × 𝑦𝐵 {𝑦}) = 𝑦𝐵 (𝐴 × {𝑦})
42, 3eqtr3i 2849 . 2 (𝐴 × 𝐵) = 𝑦𝐵 (𝐴 × {𝑦})
5 id 22 . . 3 (𝐵𝑊𝐵𝑊)
6 fconstmpt 5601 . . . . 5 (𝐴 × {𝑦}) = (𝑥𝐴𝑦)
7 mptexg 6975 . . . . 5 (𝐴𝑉 → (𝑥𝐴𝑦) ∈ V)
86, 7eqeltrid 2920 . . . 4 (𝐴𝑉 → (𝐴 × {𝑦}) ∈ V)
98ralrimivw 3178 . . 3 (𝐴𝑉 → ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
10 iunexg 7659 . . 3 ((𝐵𝑊 ∧ ∀𝑦𝐵 (𝐴 × {𝑦}) ∈ V) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
115, 9, 10syl2anr 599 . 2 ((𝐴𝑉𝐵𝑊) → 𝑦𝐵 (𝐴 × {𝑦}) ∈ V)
124, 11eqeltrid 2920 1 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115  ∀wral 3133  Vcvv 3480  {csn 4550  ∪ ciun 4905   ↦ cmpt 5132   × cxp 5540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator