Theorem xpsfval 17508

Description: The value of the function appearing in xpsval 17512. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypothesis

Ref	Expression
xpsff1o.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})

Assertion

Ref	Expression
xpsfval	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (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

Step	Hyp	Ref	Expression
1		simpl 483	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
2	1	opeq2d 4879	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3		simpr 485	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
4	3	opeq2d 4879	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
5	2, 4	preq12d 4744	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
6		xpsff1o.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
7		prex 5431	. 2 ⊢ {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V
8	5, 6, 7	ovmpoa 7559	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∅c0 4321  {cpr 4629  ⟨cop 4633  (class class class)co 7405   ∈ cmpo 7407  1_oc1o 8455

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410

This theorem is referenced by:  xpsff1o  17509  xpsaddlem  17515  xpsvsca  17519  xpsle  17521  xpsdsval  23878