Theorem fpar 8141

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 30481. (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 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))

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

Step	Hyp	Ref	Expression
1		fparlem3 8139	. . 3 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))
2		fparlem4 8140	. . 3 ⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))
3	1, 2	ineqan12d 4222	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V))))) = (∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))))
4		fpar.1	. 2 ⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
5		opex 5469	. . . 4 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
6	5	dfmpo 8127	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
7		inxp 5842	. . . . . . . 8 ⊢ ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)})))
8		inxp 5842	. . . . . . . . . 10 ⊢ (({𝑥} × V) ∩ (V × {𝑦})) = (({𝑥} ∩ V) × (V ∩ {𝑦}))
9		inv1 4398	. . . . . . . . . . 11 ⊢ ({𝑥} ∩ V) = {𝑥}
10		incom 4209	. . . . . . . . . . . 12 ⊢ (V ∩ {𝑦}) = ({𝑦} ∩ V)
11		inv1 4398	. . . . . . . . . . . 12 ⊢ ({𝑦} ∩ V) = {𝑦}
12	10, 11	eqtri 2765	. . . . . . . . . . 11 ⊢ (V ∩ {𝑦}) = {𝑦}
13	9, 12	xpeq12i 5713	. . . . . . . . . 10 ⊢ (({𝑥} ∩ V) × (V ∩ {𝑦})) = ({𝑥} × {𝑦})
14		vex 3484	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
15		vex 3484	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
16	14, 15	xpsn 7161	. . . . . . . . . 10 ⊢ ({𝑥} × {𝑦}) = {⟨𝑥, 𝑦⟩}
17	8, 13, 16	3eqtri 2769	. . . . . . . . 9 ⊢ (({𝑥} × V) ∩ (V × {𝑦})) = {⟨𝑥, 𝑦⟩}
18		inxp 5842	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)})) = (({(𝐹‘𝑥)} ∩ V) × (V ∩ {(𝐺‘𝑦)}))
19		inv1 4398	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑥)} ∩ V) = {(𝐹‘𝑥)}
20		incom 4209	. . . . . . . . . . . 12 ⊢ (V ∩ {(𝐺‘𝑦)}) = ({(𝐺‘𝑦)} ∩ V)
21		inv1 4398	. . . . . . . . . . . 12 ⊢ ({(𝐺‘𝑦)} ∩ V) = {(𝐺‘𝑦)}
22	20, 21	eqtri 2765	. . . . . . . . . . 11 ⊢ (V ∩ {(𝐺‘𝑦)}) = {(𝐺‘𝑦)}
23	19, 22	xpeq12i 5713	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} ∩ V) × (V ∩ {(𝐺‘𝑦)})) = ({(𝐹‘𝑥)} × {(𝐺‘𝑦)})
24		fvex 6919	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
25		fvex 6919	. . . . . . . . . . 11 ⊢ (𝐺‘𝑦) ∈ V
26	24, 25	xpsn 7161	. . . . . . . . . 10 ⊢ ({(𝐹‘𝑥)} × {(𝐺‘𝑦)}) = {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩}
27	18, 23, 26	3eqtri 2769	. . . . . . . . 9 ⊢ (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)})) = {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩}
28	17, 27	xpeq12i 5713	. . . . . . . 8 ⊢ ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)}))) = ({⟨𝑥, 𝑦⟩} × {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩})
29		opex 5469	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
30	29, 5	xpsn 7161	. . . . . . . 8 ⊢ ({⟨𝑥, 𝑦⟩} × {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩}) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
31	7, 28, 30	3eqtri 2769	. . . . . . 7 ⊢ ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
32	31	a1i 11	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩})
33	32	iuneq2i 5013	. . . . 5 ⊢ ∪ 𝑦 ∈ 𝐵 ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
34	33	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩})
35	34	iuneq2i 5013	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
36		2iunin 5076	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = (∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))
37	6, 35, 36	3eqtr2i 2771	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))
38	3, 4, 37	3eqtr4g 2802	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950  {csn 4626  ⟨cop 4632  ∪ ciun 4991   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   ∘ ccom 5689   Fn wfn 6556  ‘cfv 6561   ∈ cmpo 7433  1^st c1st 8012  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015

This theorem is referenced by:  fsplitfpar  8143  ex-fpar  30481