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Theorem fpar 7326
Description: Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as 𝑧 = ((√‘𝑥) + (abs‘𝑦)). (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 7324 . . 3 (𝐹 Fn 𝐴 → ((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)))
2 fparlem4 7325 . . 3 (𝐺 Fn 𝐵 → ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
31, 2ineqan12d 3849 . 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 4962 . . . 4 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
65dfmpt2 7312 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
7 inxp 5287 . . . . . . . 8 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})))
8 inxp 5287 . . . . . . . . . 10 (({𝑥} × V) ∩ (V × {𝑦})) = (({𝑥} ∩ V) × (V ∩ {𝑦}))
9 inv1 4003 . . . . . . . . . . 11 ({𝑥} ∩ V) = {𝑥}
10 incom 3838 . . . . . . . . . . . 12 (V ∩ {𝑦}) = ({𝑦} ∩ V)
11 inv1 4003 . . . . . . . . . . . 12 ({𝑦} ∩ V) = {𝑦}
1210, 11eqtri 2673 . . . . . . . . . . 11 (V ∩ {𝑦}) = {𝑦}
139, 12xpeq12i 5171 . . . . . . . . . 10 (({𝑥} ∩ V) × (V ∩ {𝑦})) = ({𝑥} × {𝑦})
14 vex 3234 . . . . . . . . . . 11 𝑥 ∈ V
15 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
1614, 15xpsn 6447 . . . . . . . . . 10 ({𝑥} × {𝑦}) = {⟨𝑥, 𝑦⟩}
178, 13, 163eqtri 2677 . . . . . . . . 9 (({𝑥} × V) ∩ (V × {𝑦})) = {⟨𝑥, 𝑦⟩}
18 inxp 5287 . . . . . . . . . 10 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)}))
19 inv1 4003 . . . . . . . . . . 11 ({(𝐹𝑥)} ∩ V) = {(𝐹𝑥)}
20 incom 3838 . . . . . . . . . . . 12 (V ∩ {(𝐺𝑦)}) = ({(𝐺𝑦)} ∩ V)
21 inv1 4003 . . . . . . . . . . . 12 ({(𝐺𝑦)} ∩ V) = {(𝐺𝑦)}
2220, 21eqtri 2673 . . . . . . . . . . 11 (V ∩ {(𝐺𝑦)}) = {(𝐺𝑦)}
2319, 22xpeq12i 5171 . . . . . . . . . 10 (({(𝐹𝑥)} ∩ V) × (V ∩ {(𝐺𝑦)})) = ({(𝐹𝑥)} × {(𝐺𝑦)})
24 fvex 6239 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
25 fvex 6239 . . . . . . . . . . 11 (𝐺𝑦) ∈ V
2624, 25xpsn 6447 . . . . . . . . . 10 ({(𝐹𝑥)} × {(𝐺𝑦)}) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2718, 23, 263eqtri 2677 . . . . . . . . 9 (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)})) = {⟨(𝐹𝑥), (𝐺𝑦)⟩}
2817, 27xpeq12i 5171 . . . . . . . 8 ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹𝑥)} × V) ∩ (V × {(𝐺𝑦)}))) = ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩})
29 opex 4962 . . . . . . . . 9 𝑥, 𝑦⟩ ∈ V
3029, 5xpsn 6447 . . . . . . . 8 ({⟨𝑥, 𝑦⟩} × {⟨(𝐹𝑥), (𝐺𝑦)⟩}) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
317, 28, 303eqtri 2677 . . . . . . 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 4571 . . . . 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 4571 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹𝑥), (𝐺𝑦)⟩⟩}
36 2iunin 4620 . . 3 𝑥𝐴 𝑦𝐵 ((({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺𝑦)}))) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
376, 35, 363eqtr2i 2679 . 2 (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = ( 𝑥𝐴 (({𝑥} × V) × ({(𝐹𝑥)} × V)) ∩ 𝑦𝐵 ((V × {𝑦}) × (V × {(𝐺𝑦)})))
383, 4, 373eqtr4g 2710 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → 𝐻 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  {csn 4210  cop 4216   ciun 4552   × cxp 5141  ccnv 5142  cres 5145  ccom 5147   Fn wfn 5921  cfv 5926  cmpt2 6692  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by: (None)
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