Theorem fpar 8116

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 31045. (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 8114	. . 3 ⊢ (𝐹 Fn 𝐴 → (◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))
2		fparlem4 8115	. . 3 ⊢ (𝐺 Fn 𝐵 → (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))) = ∪ 𝑦 ∈ 𝐵 ((V × {𝑦}) × (V × {(𝐺‘𝑦)})))
3	1, 2	ineqan12d 4168	. 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 5432	. . . 4 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
6	5	dfmpo 8102	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
7		inxp 5809	. . . . . . . 8 ⊢ ((({𝑥} × V) × ({(𝐹‘𝑥)} × V)) ∩ ((V × {𝑦}) × (V × {(𝐺‘𝑦)}))) = ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)})))
8		inxp 5809	. . . . . . . . . 10 ⊢ (({𝑥} × V) ∩ (V × {𝑦})) = (({𝑥} ∩ V) × (V ∩ {𝑦}))
9		inv1 4348	. . . . . . . . . . 11 ⊢ ({𝑥} ∩ V) = {𝑥}
10		incom 4155	. . . . . . . . . . . 12 ⊢ (V ∩ {𝑦}) = ({𝑦} ∩ V)
11		inv1 4348	. . . . . . . . . . . 12 ⊢ ({𝑦} ∩ V) = {𝑦}
12	10, 11	eqtri 2784	. . . . . . . . . . 11 ⊢ (V ∩ {𝑦}) = {𝑦}
13	9, 12	xpeq12i 5679	. . . . . . . . . 10 ⊢ (({𝑥} ∩ V) × (V ∩ {𝑦})) = ({𝑥} × {𝑦})
14		vex 3455	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
15		vex 3455	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
16	14, 15	xpsn 7133	. . . . . . . . . 10 ⊢ ({𝑥} × {𝑦}) = {⟨𝑥, 𝑦⟩}
17	8, 13, 16	3eqtri 2788	. . . . . . . . 9 ⊢ (({𝑥} × V) ∩ (V × {𝑦})) = {⟨𝑥, 𝑦⟩}
18		inxp 5809	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)})) = (({(𝐹‘𝑥)} ∩ V) × (V ∩ {(𝐺‘𝑦)}))
19		inv1 4348	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑥)} ∩ V) = {(𝐹‘𝑥)}
20		incom 4155	. . . . . . . . . . . 12 ⊢ (V ∩ {(𝐺‘𝑦)}) = ({(𝐺‘𝑦)} ∩ V)
21		inv1 4348	. . . . . . . . . . . 12 ⊢ ({(𝐺‘𝑦)} ∩ V) = {(𝐺‘𝑦)}
22	20, 21	eqtri 2784	. . . . . . . . . . 11 ⊢ (V ∩ {(𝐺‘𝑦)}) = {(𝐺‘𝑦)}
23	19, 22	xpeq12i 5679	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} ∩ V) × (V ∩ {(𝐺‘𝑦)})) = ({(𝐹‘𝑥)} × {(𝐺‘𝑦)})
24		fvex 6890	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
25		fvex 6890	. . . . . . . . . . 11 ⊢ (𝐺‘𝑦) ∈ V
26	24, 25	xpsn 7133	. . . . . . . . . 10 ⊢ ({(𝐹‘𝑥)} × {(𝐺‘𝑦)}) = {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩}
27	18, 23, 26	3eqtri 2788	. . . . . . . . 9 ⊢ (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)})) = {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩}
28	17, 27	xpeq12i 5679	. . . . . . . 8 ⊢ ((({𝑥} × V) ∩ (V × {𝑦})) × (({(𝐹‘𝑥)} × V) ∩ (V × {(𝐺‘𝑦)}))) = ({⟨𝑥, 𝑦⟩} × {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩})
29		opex 5432	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
30	29, 5	xpsn 7133	. . . . . . . 8 ⊢ ({⟨𝑥, 𝑦⟩} × {⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩}) = {⟨⟨𝑥, 𝑦⟩, ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩⟩}
31	7, 28, 30	3eqtri 2788	. . . . . . 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 4973	. . . . 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 4973	. . 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 5036	. . 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 2790	. 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 2821	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  {csn 4584  ⟨cop 4590  ∪ ciun 4951   × cxp 5649  ^◡ccnv 5650   ↾ cres 5653   ∘ ccom 5655   Fn wfn 6526  ‘cfv 6531   ∈ cmpo 7414  1^st c1st 7988  2^nd c2nd 7989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991

This theorem is used by:  fsplitfpar  8118  ex-fpar  31045