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Theorem xpsfval 16835
Description: The value of the function appearing in xpsval 16839. (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 486 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
21opeq2d 4796 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3 simpr 488 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
43opeq2d 4796 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
52, 4preq12d 4661 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
6 xpsff1o.f . 2 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
7 prex 5320 . 2 {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V
85, 6, 7ovmpoa 7294 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  c0 4275  {cpr 4551  cop 4555  (class class class)co 7145  cmpo 7147  1oc1o 8085
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-sep 5189  ax-nul 5196  ax-pr 5317
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-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150
This theorem is referenced by:  xpsff1o  16836  xpsaddlem  16842  xpsvsca  16846  xpsle  16848  xpsdsval  22984
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