Theorem xpsfval 17521

Description: The value of the function appearing in xpsval 17525. (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 4824	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
3		simpr 484	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
4	3	opeq2d 4824	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
5	2, 4	preq12d 4686	. 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 5375	. 2 ⊢ {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V
8	5, 6, 7	ovmpoa 7515	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∅c0 4274  {cpr 4570  ⟨cop 4574  (class class class)co 7360   ∈ cmpo 7362  1_oc1o 8391

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  xpsff1o  17522  xpsaddlem  17528  xpsvsca  17532  xpsle  17534  xpsdsval  24356