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Theorem fpar 7849
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 28411. (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 7847 . . 3 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))
2 fparlem4 7848 . . 3 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
31, 2ineqan12d 4115 . 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 5332 . . . 4 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
65dfmpo 7835 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
7 inxp 5685 . . . . . . . 8 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})))
8 inxp 5685 . . . . . . . . . 10 (({𝑥} × V) ∩ (V × {𝑦})) = (({𝑥} ∩ V) × (V ∩ {𝑦}))
9 inv1 4293 . . . . . . . . . . 11 ({𝑥} ∩ V) = {𝑥}
10 incom 4101 . . . . . . . . . . . 12 (V ∩ {𝑦}) = ({𝑦} ∩ V)
11 inv1 4293 . . . . . . . . . . . 12 ({𝑦} ∩ V) = {𝑦}
1210, 11eqtri 2762 . . . . . . . . . . 11 (V ∩ {𝑦}) = {𝑦}
139, 12xpeq12i 5563 . . . . . . . . . 10 (({𝑥} ∩ V) × (V ∩ {𝑦})) = ({𝑥} × {𝑦})
14 vex 3404 . . . . . . . . . . 11 𝑥 ∈ V
15 vex 3404 . . . . . . . . . . 11 𝑦 ∈ V
1614, 15xpsn 6925 . . . . . . . . . 10 ({𝑥} × {𝑦}) = {⟨𝑥, 𝑦⟩}
178, 13, 163eqtri 2766 . . . . . . . . 9 (({𝑥} × V) ∩ (V × {𝑦})) = {⟨𝑥, 𝑦⟩}
18 inxp 5685 . . . . . . . . . 10 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)}))
19 inv1 4293 . . . . . . . . . . 11 ({(𝐹𝑥)} ∩ V) = {(𝐹𝑥)}
20 incom 4101 . . . . . . . . . . . 12 (V ∩ {(𝐺𝑦)}) = ({(𝐺𝑦)} ∩ V)
21 inv1 4293 . . . . . . . . . . . 12 ({(𝐺𝑦)} ∩ V) = {(𝐺𝑦)}
2220, 21eqtri 2762 . . . . . . . . . . 11 (V ∩ {(𝐺𝑦)}) = {(𝐺𝑦)}
2319, 22xpeq12i 5563 . . . . . . . . . 10 (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)})) = ({(𝐹𝑥)} × {(𝐺𝑦)})
24 fvex 6699 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
25 fvex 6699 . . . . . . . . . . 11 (𝐺𝑦) ∈ V
2624, 25xpsn 6925 . . . . . . . . . 10 ({(𝐹𝑥)} × {(𝐺𝑦)}) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2718, 23, 263eqtri 2766 . . . . . . . . 9 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2817, 27xpeq12i 5563 . . . . . . . 8 ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)}))) = ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩})
29 opex 5332 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3029, 5xpsn 6925 . . . . . . . 8 ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩}) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
317, 28, 303eqtri 2766 . . . . . . 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 4912 . . . . 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 4912 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
36 2iunin 4971 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
376, 35, 363eqtr2i 2768 . 2 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
383, 4, 373eqtr4g 2799 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3400  cin 3852  {csn 4526  cop 4532   ciun 4891   × cxp 5533  ccnv 5534  cres 5537  ccom 5539   Fn wfn 6344  cfv 6349  cmpo 7184  1st c1st 7724  2nd c2nd 7725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727
This theorem is referenced by:  fsplitfpar  7852  ex-fpar  28411
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