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Theorem xpsfval 17449
Description: The value of the function appearing in xpsval 17453. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
Assertion
Ref Expression
xpsfval ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem xpsfval
StepHypRef Expression
1 simpl 484 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
21opeq2d 4838 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3 simpr 486 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
43opeq2d 4838 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
52, 4preq12d 4703 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
6 xpsff1o.f . 2 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
7 prex 5390 . 2 {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V
85, 6, 7ovmpoa 7511 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  c0 4283  {cpr 4589  cop 4593  (class class class)co 7358  cmpo 7360  1oc1o 8406
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-sep 5257  ax-nul 5264  ax-pr 5385
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-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  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-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  xpsff1o  17450  xpsaddlem  17456  xpsvsca  17460  xpsle  17462  xpsdsval  23737
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