Theorem xpsfval 17194

Description: The value of the function appearing in xpsval 17198. (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 482	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
2	1	opeq2d 4808	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3		simpr 484	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
4	3	opeq2d 4808	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
5	2, 4	preq12d 4674	. 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 5350	. 2 ⊢ {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V
8	5, 6, 7	ovmpoa 7406	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∅c0 4253  {cpr 4560  ⟨cop 4564  (class class class)co 7255   ∈ cmpo 7257  1_oc1o 8260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260

This theorem is referenced by:  xpsff1o  17195  xpsaddlem  17201  xpsvsca  17205  xpsle  17207  xpsdsval  23442