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Theorem fsplitfpar 8101

Description: Merge two functions with a common argument in parallel. Combination of fsplit 8100 and fpar 8099. (Contributed by AV, 3-Jan-2024.)

**Hypotheses**
Ref	Expression
fsplitfpar.h	⊢ 𝐻 = ((^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) ∩ (^◡(2^nd ↾ (V × V)) ∘ (𝐺 ∘ (2^nd ↾ (V × V)))))
fsplitfpar.s	⊢ 𝑆 = (^◡(1^st ↾ I ) ↾ 𝐴)

**Assertion**
Ref	Expression
fsplitfpar	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹 𝑥,𝐺 Allowed substitution hints: 𝑆(𝑥) 𝐻(𝑥)

Proof of Theorem fsplitfpar Dummy variables 𝑎 𝑝 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsplitfpar.s	. . . . . . . . . 10 ⊢ 𝑆 = (^◡(1^st ↾ I ) ↾ 𝐴)
2		fsplit 8100	. . . . . . . . . . 11 ⊢ ^◡(1^st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
3	2	reseq1i 5964	. . . . . . . . . 10 ⊢ (^◡(1^st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
4	1, 3	eqtri 2788	. . . . . . . . 9 ⊢ 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
5	4	fveq1i 6872	. . . . . . . 8 ⊢ (𝑆‘𝑎) = (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎)
6	5	a1i 11	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎))
7		fvres 6890	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎))
8		eqidd 2766	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩))
9		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
10	9, 9	opeq12d 4841	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
11	10	adantl 486	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
12		elex 3478	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ V)
13		opex 5435	. . . . . . . . . . 11 ⊢ ⟨𝑎, 𝑎⟩ ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → ⟨𝑎, 𝑎⟩ ∈ V)
15	8, 11, 12, 14	fvmptd 6987	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩)
16	7, 15	eqtrd 2800	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
17	16	adantl 486	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
18	6, 17	eqtrd 2800	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = ⟨𝑎, 𝑎⟩)
19	18	fveq2d 6875	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩))
20		df-ov 7403	. . . . . 6 ⊢ (𝑎𝐻𝑎) = (𝐻‘⟨𝑎, 𝑎⟩)
21		fsplitfpar.h	. . . . . . . . 9 ⊢ 𝐻 = ((^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) ∩ (^◡(2^nd ↾ (V × V)) ∘ (𝐺 ∘ (2^nd ↾ (V × V)))))
22	21	fpar 8099	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
23	22	adantr 485	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
24		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
25	24	adantr 485	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹‘𝑥) = (𝐹‘𝑎))
26		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎))
27	26	adantl 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐺‘𝑦) = (𝐺‘𝑎))
28	25, 27	opeq12d 4841	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
29	28	adantl 486	. . . . . . 7 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑎)) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
30		simpr 489	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
31		opex 5435	. . . . . . . 8 ⊢ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V
32	31	a1i 11	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V)
33	23, 29, 30, 30, 32	ovmpod 7552	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
34	20, 33	eqtr3id 2814	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
35	19, 34	eqtrd 2800	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
36		eqid 2765	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
37	36	fnmpt 6665	. . . . . . . . 9 ⊢ (∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
38	13	a1i 11	. . . . . . . . 9 ⊢ (𝑎 ∈ V → ⟨𝑎, 𝑎⟩ ∈ V)
39	37, 38	mprg 3085	. . . . . . . 8 ⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V
40		ssv 3963	. . . . . . . 8 ⊢ 𝐴 ⊆ V
41		fnssres 6648	. . . . . . . 8 ⊢ (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
42	39, 40, 41	mp2an 704	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴
43		fsplit 8100	. . . . . . . . . 10 ⊢ ^◡(1^st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
44	43	reseq1i 5964	. . . . . . . . 9 ⊢ (^◡(1^st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
45	1, 44	eqtri 2788	. . . . . . . 8 ⊢ 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
46	45	fneq1i 6622	. . . . . . 7 ⊢ (𝑆 Fn 𝐴 ↔ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
47	42, 46	mpbir 234	. . . . . 6 ⊢ 𝑆 Fn 𝐴
48	47	a1i 11	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
49		fvco2 6968	. . . . 5 ⊢ ((𝑆 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎)))
50	48, 49	sylan 591	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎)))
51		fveq2 6871	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐺‘𝑥) = (𝐺‘𝑎))
52	24, 51	opeq12d 4841	. . . . . 6 ⊢ (𝑥 = 𝑎 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
53		eqid 2765	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
54	52, 53, 31	fvmpt 6979	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
55	54	adantl 486	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
56	35, 50, 55	3eqtr4d 2810	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))
57	56	ralrimiva 3157	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))
58		opex 5435	. . . . . . . 8 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V
59	58	a1i 11	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V)
60	59	ralrimivva 3208	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V)
61		eqid 2765	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
62	61	fnmpo 8054	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴))
63	60, 62	syl 18	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴))
64	22	fneq1d 6618	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 Fn (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴)))
65	63, 64	mpbird 260	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 Fn (𝐴 × 𝐴))
66	13	a1i 11	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ V) → ⟨𝑎, 𝑎⟩ ∈ V)
67	66	ralrimiva 3157	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V)
68	67, 37	syl 18	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
69	68, 40, 41	sylancl 597	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
70	69, 46	sylibr 237	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
71	45	rneqi 5917	. . . . . 6 ⊢ ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
72		mptima 6064	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
73		df-ima 5664	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
74		eqid 2765	. . . . . . . 8 ⊢ (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
75	74	rnmpt 5937	. . . . . . 7 ⊢ ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
76	72, 73, 75	3eqtr3i 2796	. . . . . 6 ⊢ ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
77	71, 76	eqtri 2788	. . . . 5 ⊢ ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
78		elinel2 4157	. . . . . . . . 9 ⊢ (𝑎 ∈ (V ∩ 𝐴) → 𝑎 ∈ 𝐴)
79		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎 ∈ 𝐴)
80	79, 79	opelxpd 5690	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))
81		eleq1 2853	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
82	81	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
83	80, 82	mpbird 260	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑝 ∈ (𝐴 × 𝐴))
84	83	ex 417	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
85	78, 84	syl 18	. . . . . . . 8 ⊢ (𝑎 ∈ (V ∩ 𝐴) → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
86	85	rexlimiv 3159	. . . . . . 7 ⊢ (∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))
87	86	abssi 4024	. . . . . 6 ⊢ {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} ⊆ (𝐴 × 𝐴)
88	87	a1i 11	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} ⊆ (𝐴 × 𝐴))
89	77, 88	eqsstrid 3977	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴))
90		fnco 6643	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻 ∘ 𝑆) Fn 𝐴)
91	65, 70, 89, 90	syl3anc 1394	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) Fn 𝐴)
92		opex 5435	. . . . . 6 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V
93	92	a1i 11	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V)
94	93	ralrimiva 3157	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V)
95	53	fnmpt 6665	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴)
96	94, 95	syl 18	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴)
97		eqfnfv 7015	. . 3 ⊢ (((𝐻 ∘ 𝑆) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴) → ((𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ↔ ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎)))
98	91, 96, 97	syl2anc 595	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) ↔ ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎)))
99	57, 98	mpbird 260	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 {cab 2743 ∀wral 3079 ∃wrex 3089 Vcvv 3457 ∩ cin 3906 ⊆ wss 3907 ⟨cop 4591 ↦ cmpt 5185 I cid 5545 × cxp 5649 ^◡ccnv 5650 ran crn 5652 ↾ cres 5653 “ cima 5654 ∘ ccom 5655 Fn wfn 6520 ‘cfv 6525 (class class class)co 7400 ∈ cmpo 7402 1^st c1st 7972 2^nd c2nd 7973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pr 5394 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 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 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7974 df-2nd 7975

This theorem is referenced by: offsplitfpar 8102

W3C validator