Theorem xpsfval 13652

Description: The value of the function appearing in xpsval 14184. (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		0lt2o 6708	. . . 4 ⊢ ∅ ∈ 2o
2		simpl 109	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
3		opexg 4366	. . . 4 ⊢ ((∅ ∈ 2o ∧ 𝑋 ∈ 𝐴) → ⟨∅, 𝑋⟩ ∈ V)
4	1, 2, 3	sylancr 418	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨∅, 𝑋⟩ ∈ V)
5		1lt2o 6709	. . . 4 ⊢ 1o ∈ 2o
6		simpr 110	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
7		opexg 4366	. . . 4 ⊢ ((1o ∈ 2o ∧ 𝑌 ∈ 𝐵) → ⟨1o, 𝑌⟩ ∈ V)
8	5, 6, 7	sylancr 418	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨1o, 𝑌⟩ ∈ V)
9		prexg 4347	. . 3 ⊢ ((⟨∅, 𝑋⟩ ∈ V ∧ ⟨1o, 𝑌⟩ ∈ V) → {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V)
10	4, 8, 9	syl2anc 415	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V)
11		simpl 109	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
12	11	opeq2d 3909	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
13		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
14	13	opeq2d 3909	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
15	12, 14	preq12d 3795	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
16		xpsff1o.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
17	15, 16	ovmpoga 6212	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
18	10, 17	mpd3an3 1379	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  {cpr 3709  ⟨cop 3711  (class class class)co 6079   ∈ cmpo 6081  1_oc1o 6674  2_oc2o 6675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1o 6681  df-2o 6682

This theorem is referenced by:  xpsff1o  13653