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Theorem fsplitfpar 8122

Description: Merge two functions with a common argument in parallel. Combination of fsplit 8121 and fpar 8120. (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 8121	. . . . . . . . . . 11 ⊢ ^◡(1^st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
3	2	reseq1i 5979	. . . . . . . . . 10 ⊢ (^◡(1^st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
4	1, 3	eqtri 2789	. . . . . . . . 9 ⊢ 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
5	4	fveq1i 6889	. . . . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎))
8		eqidd 2767	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩))
9		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
10	9, 9	opeq12d 4851	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
11	10	adantl 487	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
12		elex 3479	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ V)
13		opex 5450	. . . . . . . . . . 11 ⊢ ⟨𝑎, 𝑎⟩ ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐴 → ⟨𝑎, 𝑎⟩ ∈ V)
15	8, 11, 12, 14	fvmptd 7004	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩)
16	7, 15	eqtrd 2801	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
17	16	adantl 487	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
18	6, 17	eqtrd 2801	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑆‘𝑎) = ⟨𝑎, 𝑎⟩)
19	18	fveq2d 6892	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩))
20		df-ov 7426	. . . . . 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 8120	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
23	22	adantr 486	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩))
24		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
25	24	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹‘𝑥) = (𝐹‘𝑎))
26		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑦) = (𝐺‘𝑎))
27	26	adantl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐺‘𝑦) = (𝐺‘𝑎))
28	25, 27	opeq12d 4851	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
29	28	adantl 487	. . . . . . 7 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑎)) → ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
30		simpr 490	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
31		opex 5450	. . . . . . . 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 7575	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
34	20, 33	eqtr3id 2815	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
35	19, 34	eqtrd 2801	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝐻‘(𝑆‘𝑎)) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
36		eqid 2766	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
37	36	fnmpt 6682	. . . . . . . . 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 3088	. . . . . . . 8 ⊢ (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V
40		ssv 3964	. . . . . . . 8 ⊢ 𝐴 ⊆ V
41		fnssres 6665	. . . . . . . 8 ⊢ (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
42	39, 40, 41	mp2an 705	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴
43		fsplit 8121	. . . . . . . . . 10 ⊢ ^◡(1^st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
44	43	reseq1i 5979	. . . . . . . . 9 ⊢ (^◡(1^st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
45	1, 44	eqtri 2789	. . . . . . . 8 ⊢ 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
46	45	fneq1i 6639	. . . . . . 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 6985	. . . . 5 ⊢ ((𝑆 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎)))
50	48, 49	sylan 592	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = (𝐻‘(𝑆‘𝑎)))
51		fveq2 6888	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐺‘𝑥) = (𝐺‘𝑎))
52	24, 51	opeq12d 4851	. . . . . 6 ⊢ (𝑥 = 𝑎 → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
53		eqid 2766	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
54	52, 53, 31	fvmpt 6996	. . . . 5 ⊢ (𝑎 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
55	54	adantl 487	. . . 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 2811	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))
57	56	ralrimiva 3160	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑎 ∈ 𝐴 ((𝐻 ∘ 𝑆)‘𝑎) = ((𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)‘𝑎))
58		opex 5450	. . . . . . . 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 3211	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩ ∈ V)
61		eqid 2766	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
62	61	fnmpo 8075	. . . . . 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 6635	. . . . 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 3160	. . . . . . 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 598	. . . . 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 5932	. . . . . 6 ⊢ ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
72		mptima 6079	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
73		df-ima 5679	. . . . . . 7 ⊢ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
74		eqid 2766	. . . . . . . 8 ⊢ (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
75	74	rnmpt 5952	. . . . . . 7 ⊢ ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
76	72, 73, 75	3eqtr3i 2797	. . . . . 6 ⊢ ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
77	71, 76	eqtri 2789	. . . . 5 ⊢ ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
78		elinel2 4158	. . . . . . . . 9 ⊢ (𝑎 ∈ (V ∩ 𝐴) → 𝑎 ∈ 𝐴)
79		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎 ∈ 𝐴)
80	79, 79	opelxpd 5705	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))
81		eleq1 2854	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
82	81	adantl 487	. . . . . . . . . . 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 418	. . . . . . . . 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 3162	. . . . . . 7 ⊢ (∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))
87	86	abssi 4025	. . . . . 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 3978	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴))
90		fnco 6660	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻 ∘ 𝑆) Fn 𝐴)
91	65, 70, 89, 90	syl3anc 1398	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) Fn 𝐴)
92		opex 5450	. . . . . 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 3160	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ∀𝑥 ∈ 𝐴 ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V)
95	53	fnmpt 6682	. . . 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 7032	. . 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 596	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {cab 2744 ∀wral 3082 ∃wrex 3092 Vcvv 3458 ∩ cin 3907 ⊆ wss 3908 ⟨cop 4600 ↦ cmpt 5197 I cid 5560 × cxp 5664 ^◡ccnv 5665 ran crn 5667 ↾ cres 5668 “ cima 5669 ∘ ccom 5670 Fn wfn 6538 ‘cfv 6543 (class class class)co 7423 ∈ cmpo 7425 1^st c1st 7993 2^nd c2nd 7994

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pr 5409 ax-un 7745

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 7995 df-2nd 7996

This theorem is used by: offsplitfpar 8123

W3C validator