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Theorem fpar 7800
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 28168. (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 7798 . . 3 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))
2 fparlem4 7799 . . 3 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
31, 2ineqan12d 4188 . 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 5347 . . . 4 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
65dfmpo 7786 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
7 inxp 5696 . . . . . . . 8 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})))
8 inxp 5696 . . . . . . . . . 10 (({𝑥} × V) ∩ (V × {𝑦})) = (({𝑥} ∩ V) × (V ∩ {𝑦}))
9 inv1 4345 . . . . . . . . . . 11 ({𝑥} ∩ V) = {𝑥}
10 incom 4175 . . . . . . . . . . . 12 (V ∩ {𝑦}) = ({𝑦} ∩ V)
11 inv1 4345 . . . . . . . . . . . 12 ({𝑦} ∩ V) = {𝑦}
1210, 11eqtri 2841 . . . . . . . . . . 11 (V ∩ {𝑦}) = {𝑦}
139, 12xpeq12i 5576 . . . . . . . . . 10 (({𝑥} ∩ V) × (V ∩ {𝑦})) = ({𝑥} × {𝑦})
14 vex 3495 . . . . . . . . . . 11 𝑥 ∈ V
15 vex 3495 . . . . . . . . . . 11 𝑦 ∈ V
1614, 15xpsn 6895 . . . . . . . . . 10 ({𝑥} × {𝑦}) = {⟨𝑥, 𝑦⟩}
178, 13, 163eqtri 2845 . . . . . . . . 9 (({𝑥} × V) ∩ (V × {𝑦})) = {⟨𝑥, 𝑦⟩}
18 inxp 5696 . . . . . . . . . 10 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)}))
19 inv1 4345 . . . . . . . . . . 11 ({(𝐹𝑥)} ∩ V) = {(𝐹𝑥)}
20 incom 4175 . . . . . . . . . . . 12 (V ∩ {(𝐺𝑦)}) = ({(𝐺𝑦)} ∩ V)
21 inv1 4345 . . . . . . . . . . . 12 ({(𝐺𝑦)} ∩ V) = {(𝐺𝑦)}
2220, 21eqtri 2841 . . . . . . . . . . 11 (V ∩ {(𝐺𝑦)}) = {(𝐺𝑦)}
2319, 22xpeq12i 5576 . . . . . . . . . 10 (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)})) = ({(𝐹𝑥)} × {(𝐺𝑦)})
24 fvex 6676 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
25 fvex 6676 . . . . . . . . . . 11 (𝐺𝑦) ∈ V
2624, 25xpsn 6895 . . . . . . . . . 10 ({(𝐹𝑥)} × {(𝐺𝑦)}) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2718, 23, 263eqtri 2845 . . . . . . . . 9 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2817, 27xpeq12i 5576 . . . . . . . 8 ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)}))) = ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩})
29 opex 5347 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3029, 5xpsn 6895 . . . . . . . 8 ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩}) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
317, 28, 303eqtri 2845 . . . . . . 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 4931 . . . . 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 4931 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
36 2iunin 4989 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
376, 35, 363eqtr2i 2847 . 2 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
383, 4, 373eqtr4g 2878 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  {csn 4557  cop 4563   ciun 4910   × cxp 5546  ccnv 5547  cres 5550  ccom 5552   Fn wfn 6343  cfv 6348  cmpo 7147  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679
This theorem is referenced by:  fsplitfpar  7803  ex-fpar  28168
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