Theorem xpsfval 13550

Description: The value of the function appearing in xpsval 13554. (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 6673	. . . 4 ⊢ ∅ ∈ 2o
2		simpl 109	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
3		opexg 4343	. . . 4 ⊢ ((∅ ∈ 2o ∧ 𝑋 ∈ 𝐴) → ⟨∅, 𝑋⟩ ∈ V)
4	1, 2, 3	sylancr 414	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨∅, 𝑋⟩ ∈ V)
5		1lt2o 6674	. . . 4 ⊢ 1o ∈ 2o
6		simpr 110	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
7		opexg 4343	. . . 4 ⊢ ((1o ∈ 2o ∧ 𝑌 ∈ 𝐵) → ⟨1o, 𝑌⟩ ∈ V)
8	5, 6, 7	sylancr 414	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨1o, 𝑌⟩ ∈ V)
9		prexg 4324	. . 3 ⊢ ((⟨∅, 𝑋⟩ ∈ V ∧ ⟨1o, 𝑌⟩ ∈ V) → {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V)
10	4, 8, 9	syl2anc 411	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V)
11		simpl 109	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
12	11	opeq2d 3889	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑋⟩)
13		simpr 110	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
14	13	opeq2d 3889	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ⟨1o, 𝑦⟩ = ⟨1o, 𝑌⟩)
15	12, 14	preq12d 3775	. . 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 6182	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩} ∈ V) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})
18	10, 17	mpd3an3 1375	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {⟨∅, 𝑋⟩, ⟨1o, 𝑌⟩})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2812  ∅c0 3507  {cpr 3689  ⟨cop 3691  (class class class)co 6049   ∈ cmpo 6051  1_oc1o 6639  2_oc2o 6640

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1o 6646  df-2o 6647

This theorem is referenced by:  xpsff1o  13551