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Theorem fsplitfpar 8100

Description: Merge two functions with a common argument in parallel. Combination of fsplit 8099 and fpar 8098. (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 8099	. . . . . . . . . . 11 ⊢ ^◡(1^st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
3	2	reseq1i 5949	. . . . . . . . . 10 ⊢ (^◡(1^st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
4	1, 3	eqtri 2753	. . . . . . . . 9 ⊢ 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
5	4	fveq1i 6862	. . . . . . . 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 6880	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎))
8		eqidd 2731	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩))
9		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
10	9, 9	opeq12d 4848	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
11	10	adantl 481	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
12		elex 3471	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ V)
13		opex 5427	. . . . . . . . . . 11 ⊢ ⟨𝑎, 𝑎⟩ ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → ⟨𝑎, 𝑎⟩ ∈ V)
15	8, 11, 12, 14	fvmptd 6978	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩)
16	7, 15	eqtrd 2765	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
17	16	adantl 481	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
18	6, 17	eqtrd 2765	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = ⟨𝑎, 𝑎⟩)
19	18	fveq2d 6865	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩))
20		df-ov 7393	. . . . . 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 8098	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
23	22	adantr 480	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
24		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
25	24	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹‘𝑥) = (𝐹‘𝑎))
26		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎))
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐺‘𝑦) = (𝐺‘𝑎))
28	25, 27	opeq12d 4848	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
29	28	adantl 481	. . . . . . 7 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑎)) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
30		simpr 484	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
31		opex 5427	. . . . . . . 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 7544	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
34	20, 33	eqtr3id 2779	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
35	19, 34	eqtrd 2765	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
36		eqid 2730	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
37	36	fnmpt 6661	. . . . . . . . 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 3051	. . . . . . . 8 ⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V
40		ssv 3974	. . . . . . . 8 ⊢ 𝐴 ⊆ V
41		fnssres 6644	. . . . . . . 8 ⊢ (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
42	39, 40, 41	mp2an 692	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴
43		fsplit 8099	. . . . . . . . . 10 ⊢ ^◡(1^st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
44	43	reseq1i 5949	. . . . . . . . 9 ⊢ (^◡(1^st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
45	1, 44	eqtri 2753	. . . . . . . 8 ⊢ 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
46	45	fneq1i 6618	. . . . . . 7 ⊢ (𝑆 Fn 𝐴 ↔ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
47	42, 46	mpbir 231	. . . . . 6 ⊢ 𝑆 Fn 𝐴
48	47	a1i 11	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
49		fvco2 6961	. . . . 5 ⊢ ((𝑆 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎)))
50	48, 49	sylan 580	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎)))
51		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐺‘𝑥) = (𝐺‘𝑎))
52	24, 51	opeq12d 4848	. . . . . 6 ⊢ (𝑥 = 𝑎 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
53		eqid 2730	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
54	52, 53, 31	fvmpt 6971	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
55	54	adantl 481	. . . 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 2775	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))
57	56	ralrimiva 3126	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))
58		opex 5427	. . . . . . . 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 3181	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V)
61		eqid 2730	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
62	61	fnmpo 8051	. . . . . 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 17	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) Fn (𝐴 × 𝐴))
64	22	fneq1d 6614	. . . . 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 257	. . . 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 3126	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V)
68	67, 37	syl 17	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
69	68, 40, 41	sylancl 586	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
70	69, 46	sylibr 234	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
71	45	rneqi 5904	. . . . . 6 ⊢ ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
72		mptima 6046	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
73		df-ima 5654	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
74		eqid 2730	. . . . . . . 8 ⊢ (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
75	74	rnmpt 5924	. . . . . . 7 ⊢ ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
76	72, 73, 75	3eqtr3i 2761	. . . . . 6 ⊢ ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
77	71, 76	eqtri 2753	. . . . 5 ⊢ ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
78		elinel2 4168	. . . . . . . . 9 ⊢ (𝑎 ∈ (V ∩ 𝐴) → 𝑎 ∈ 𝐴)
79		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎 ∈ 𝐴)
80	79, 79	opelxpd 5680	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))
81		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
82	81	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
83	80, 82	mpbird 257	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑝 ∈ (𝐴 × 𝐴))
84	83	ex 412	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
85	78, 84	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ (V ∩ 𝐴) → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
86	85	rexlimiv 3128	. . . . . . 7 ⊢ (∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))
87	86	abssi 4036	. . . . . 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 3988	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴))
90		fnco 6639	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻 ∘ 𝑆) Fn 𝐴)
91	65, 70, 89, 90	syl3anc 1373	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) Fn 𝐴)
92		opex 5427	. . . . . 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 3126	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V)
95	53	fnmpt 6661	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴)
96	94, 95	syl 17	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) Fn 𝐴)
97		eqfnfv 7006	. . 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 584	. 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 257	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2708 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 ⟨cop 4598 ↦ cmpt 5191 I cid 5535 × cxp 5639 ^◡ccnv 5640 ran crn 5642 ↾ cres 5643 “ cima 5644 ∘ ccom 5645 Fn wfn 6509 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7969 2^nd c2nd 7970

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972

This theorem is referenced by: offsplitfpar 8101

W3C validator