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Theorem fpar 7956
Description: Merge two functions in parallel. Use as the second argument of a composition with a binary operation to build compound functions such as (𝑥 ∈ (0[,)+∞), 𝑦 ∈ ℝ ↦ ((√‘𝑥) + (sin‘𝑦))), see also ex-fpar 28826. (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
Hypothesis
Ref Expression
fpar.1 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
Assertion
Ref Expression
fpar ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦
Allowed substitution hints:   𝐻(𝑥,𝑦)

Proof of Theorem fpar
StepHypRef Expression
1 fparlem3 7954 . . 3 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))
2 fparlem4 7955 . . 3 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
31, 2ineqan12d 4148 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V))))) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)}))))
4 fpar.1 . 2 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
5 opex 5379 . . . 4 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
65dfmpo 7942 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
7 inxp 5741 . . . . . . . 8 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})))
8 inxp 5741 . . . . . . . . . 10 (({𝑥} × V) ∩ (V × {𝑦})) = (({𝑥} ∩ V) × (V ∩ {𝑦}))
9 inv1 4328 . . . . . . . . . . 11 ({𝑥} ∩ V) = {𝑥}
10 incom 4135 . . . . . . . . . . . 12 (V ∩ {𝑦}) = ({𝑦} ∩ V)
11 inv1 4328 . . . . . . . . . . . 12 ({𝑦} ∩ V) = {𝑦}
1210, 11eqtri 2766 . . . . . . . . . . 11 (V ∩ {𝑦}) = {𝑦}
139, 12xpeq12i 5617 . . . . . . . . . 10 (({𝑥} ∩ V) × (V ∩ {𝑦})) = ({𝑥} × {𝑦})
14 vex 3436 . . . . . . . . . . 11 𝑥 ∈ V
15 vex 3436 . . . . . . . . . . 11 𝑦 ∈ V
1614, 15xpsn 7013 . . . . . . . . . 10 ({𝑥} × {𝑦}) = {⟨𝑥, 𝑦⟩}
178, 13, 163eqtri 2770 . . . . . . . . 9 (({𝑥} × V) ∩ (V × {𝑦})) = {⟨𝑥, 𝑦⟩}
18 inxp 5741 . . . . . . . . . 10 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)}))
19 inv1 4328 . . . . . . . . . . 11 ({(𝐹𝑥)} ∩ V) = {(𝐹𝑥)}
20 incom 4135 . . . . . . . . . . . 12 (V ∩ {(𝐺𝑦)}) = ({(𝐺𝑦)} ∩ V)
21 inv1 4328 . . . . . . . . . . . 12 ({(𝐺𝑦)} ∩ V) = {(𝐺𝑦)}
2220, 21eqtri 2766 . . . . . . . . . . 11 (V ∩ {(𝐺𝑦)}) = {(𝐺𝑦)}
2319, 22xpeq12i 5617 . . . . . . . . . 10 (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)})) = ({(𝐹𝑥)} × {(𝐺𝑦)})
24 fvex 6787 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
25 fvex 6787 . . . . . . . . . . 11 (𝐺𝑦) ∈ V
2624, 25xpsn 7013 . . . . . . . . . 10 ({(𝐹𝑥)} × {(𝐺𝑦)}) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2718, 23, 263eqtri 2770 . . . . . . . . 9 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2817, 27xpeq12i 5617 . . . . . . . 8 ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)}))) = ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩})
29 opex 5379 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3029, 5xpsn 7013 . . . . . . . 8 ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩}) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
317, 28, 303eqtri 2770 . . . . . . 7 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
3231a1i 11 . . . . . 6 (𝑦𝐵 → ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩})
3332iuneq2i 4945 . . . . 5 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
3433a1i 11 . . . 4 (𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩})
3534iuneq2i 4945 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
36 2iunin 5005 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
376, 35, 363eqtr2i 2772 . 2 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
383, 4, 373eqtr4g 2803 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  {csn 4561  cop 4567   ciun 4924   × cxp 5587  ccnv 5588  cres 5591  ccom 5593   Fn wfn 6428  cfv 6433  cmpo 7277  1st c1st 7829  2nd c2nd 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832
This theorem is referenced by:  fsplitfpar  7959  ex-fpar  28826
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