3f1oss1 - Mathbox for Alexander van der Vekens

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Theorem 3f1oss1 47060

Description: The composition of three bijections as bijection from the image of the domain onto the image of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)

**Assertion**
Ref	Expression
3f1oss1	⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐻 ∘ 𝐺) ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))

**Proof of Theorem 3f1oss1**
Step	Hyp	Ref	Expression
1		f1ocnv 6858	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐵–1-1-onto→𝐴)
2		f1of1 6845	. . . . . . . 8 ⊢ (^◡𝐹:𝐵–1-1-onto→𝐴 → ^◡𝐹:𝐵–1-1→𝐴)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐵–1-1→𝐴)
4	3	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → ^◡𝐹:𝐵–1-1→𝐴)
5	4	adantr 480	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ^◡𝐹:𝐵–1-1→𝐴)
6		cnvimass 6098	. . . . . . . 8 ⊢ (^◡^◡𝐹 “ 𝐶) ⊆ dom ^◡𝐹
7		f1of 6846	. . . . . . . . 9 ⊢ (^◡𝐹:𝐵–1-1-onto→𝐴 → ^◡𝐹:𝐵⟶𝐴)
8		fdm 6743	. . . . . . . . . 10 ⊢ (^◡𝐹:𝐵⟶𝐴 → dom ^◡𝐹 = 𝐵)
9	8	eqcomd 2742	. . . . . . . . 9 ⊢ (^◡𝐹:𝐵⟶𝐴 → 𝐵 = dom ^◡𝐹)
10	1, 7, 9	3syl 18	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐵 = dom ^◡𝐹)
11	6, 10	sseqtrrid 4026	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡^◡𝐹 “ 𝐶) ⊆ 𝐵)
12	11	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → (^◡^◡𝐹 “ 𝐶) ⊆ 𝐵)
13	12	adantr 480	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (^◡^◡𝐹 “ 𝐶) ⊆ 𝐵)
14		f1ofn 6847	. . . . . . . . . 10 ⊢ (^◡𝐹:𝐵–1-1-onto→𝐴 → ^◡𝐹 Fn 𝐵)
15	1, 14	syl 17	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹 Fn 𝐵)
16	15	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → ^◡𝐹 Fn 𝐵)
17	16	adantr 480	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ^◡𝐹 Fn 𝐵)
18		eqidd 2737	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (ran ^◡𝐹 ∩ 𝐶) = (ran ^◡𝐹 ∩ 𝐶))
19		eqidd 2737	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (^◡^◡𝐹 “ 𝐶) = (^◡^◡𝐹 “ 𝐶))
20	17, 18, 19	rescnvimafod 7091	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)–onto→(ran ^◡𝐹 ∩ 𝐶))
21		fof 6818	. . . . . 6 ⊢ ((^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)–onto→(ran ^◡𝐹 ∩ 𝐶) → (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)⟶(ran ^◡𝐹 ∩ 𝐶))
22	20, 21	syl 17	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)⟶(ran ^◡𝐹 ∩ 𝐶))
23		f1resf1 6810	. . . . 5 ⊢ ((^◡𝐹:𝐵–1-1→𝐴 ∧ (^◡^◡𝐹 “ 𝐶) ⊆ 𝐵 ∧ (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)⟶(ran ^◡𝐹 ∩ 𝐶)) → (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)–1-1→(ran ^◡𝐹 ∩ 𝐶))
24	5, 13, 22, 23	syl3anc 1373	. . . 4 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)–1-1→(ran ^◡𝐹 ∩ 𝐶))
25		f1of1 6845	. . . . . . . 8 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1→𝐷)
26	25	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → 𝐺:𝐶–1-1→𝐷)
27	26	adantr 480	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → 𝐺:𝐶–1-1→𝐷)
28		inss2 4237	. . . . . 6 ⊢ (ran ^◡𝐹 ∩ 𝐶) ⊆ 𝐶
29		f1ores 6860	. . . . . 6 ⊢ ((𝐺:𝐶–1-1→𝐷 ∧ (ran ^◡𝐹 ∩ 𝐶) ⊆ 𝐶) → (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→(𝐺 “ (ran ^◡𝐹 ∩ 𝐶)))
30	27, 28, 29	sylancl 586	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→(𝐺 “ (ran ^◡𝐹 ∩ 𝐶)))
31		f1ofo 6853	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹:𝐵–1-1-onto→𝐴 → ^◡𝐹:𝐵–onto→𝐴)
32		forn 6821	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹:𝐵–onto→𝐴 → ran ^◡𝐹 = 𝐴)
33	1, 31, 32	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ran ^◡𝐹 = 𝐴)
34	33	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → ran ^◡𝐹 = 𝐴)
35	34	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ran ^◡𝐹 = 𝐴)
36	35	ineq1d 4218	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (ran ^◡𝐹 ∩ 𝐶) = (𝐴 ∩ 𝐶))
37		incom 4208	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
38		dfss2 3968	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∩ 𝐴) = 𝐶)
39	38	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ 𝐴 → (𝐶 ∩ 𝐴) = 𝐶)
40	37, 39	eqtrid 2788	. . . . . . . . . . 11 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶)
41	40	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐴 ∩ 𝐶) = 𝐶)
42	36, 41	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (ran ^◡𝐹 ∩ 𝐶) = 𝐶)
43	42	imaeq2d 6076	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐺 “ (ran ^◡𝐹 ∩ 𝐶)) = (𝐺 “ 𝐶))
44		f1ofn 6847	. . . . . . . . . . . 12 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺 Fn 𝐶)
45		fnima 6696	. . . . . . . . . . . 12 ⊢ (𝐺 Fn 𝐶 → (𝐺 “ 𝐶) = ran 𝐺)
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → (𝐺 “ 𝐶) = ran 𝐺)
47		f1ofo 6853	. . . . . . . . . . . 12 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–onto→𝐷)
48		forn 6821	. . . . . . . . . . . 12 ⊢ (𝐺:𝐶–onto→𝐷 → ran 𝐺 = 𝐷)
49	47, 48	syl 17	. . . . . . . . . . 11 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → ran 𝐺 = 𝐷)
50	46, 49	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → (𝐺 “ 𝐶) = 𝐷)
51	50	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → (𝐺 “ 𝐶) = 𝐷)
52	51	adantr 480	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐺 “ 𝐶) = 𝐷)
53	43, 52	eqtrd 2776	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐺 “ (ran ^◡𝐹 ∩ 𝐶)) = 𝐷)
54	53	eqcomd 2742	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → 𝐷 = (𝐺 “ (ran ^◡𝐹 ∩ 𝐶)))
55	54	f1oeq3d 6843	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→𝐷 ↔ (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→(𝐺 “ (ran ^◡𝐹 ∩ 𝐶))))
56	30, 55	mpbird 257	. . . 4 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→𝐷)
57		f1orel 6849	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
58	57	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → Rel 𝐹)
59	58	adantr 480	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → Rel 𝐹)
60		dfrel2 6207	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
61	59, 60	sylib 218	. . . . . . . 8 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ^◡^◡𝐹 = 𝐹)
62	61	eqcomd 2742	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → 𝐹 = ^◡^◡𝐹)
63	62	imaeq1d 6075	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐹 “ 𝐶) = (^◡^◡𝐹 “ 𝐶))
64	63	f1oeq2d 6842	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐺 ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→𝐷 ↔ (𝐺 ∘ ^◡𝐹):(^◡^◡𝐹 “ 𝐶)–1-1-onto→𝐷))
65	1, 7	syl 17	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐵⟶𝐴)
66	65	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → ^◡𝐹:𝐵⟶𝐴)
67	66	adantr 480	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ^◡𝐹:𝐵⟶𝐴)
68		eqid 2736	. . . . . 6 ⊢ (ran ^◡𝐹 ∩ 𝐶) = (ran ^◡𝐹 ∩ 𝐶)
69		eqid 2736	. . . . . 6 ⊢ (^◡^◡𝐹 “ 𝐶) = (^◡^◡𝐹 “ 𝐶)
70		eqid 2736	. . . . . 6 ⊢ (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)) = (^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶))
71		f1of 6846	. . . . . . . 8 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶⟶𝐷)
72	71	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → 𝐺:𝐶⟶𝐷)
73	72	adantr 480	. . . . . 6 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → 𝐺:𝐶⟶𝐷)
74		eqid 2736	. . . . . 6 ⊢ (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)) = (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶))
75	67, 68, 69, 70, 73, 74	fcoresf1ob 47058	. . . . 5 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐺 ∘ ^◡𝐹):(^◡^◡𝐹 “ 𝐶)–1-1-onto→𝐷 ↔ ((^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)–1-1→(ran ^◡𝐹 ∩ 𝐶) ∧ (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→𝐷)))
76	64, 75	bitrd 279	. . . 4 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐺 ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→𝐷 ↔ ((^◡𝐹 ↾ (^◡^◡𝐹 “ 𝐶)):(^◡^◡𝐹 “ 𝐶)–1-1→(ran ^◡𝐹 ∩ 𝐶) ∧ (𝐺 ↾ (ran ^◡𝐹 ∩ 𝐶)):(ran ^◡𝐹 ∩ 𝐶)–1-1-onto→𝐷)))
77	24, 56, 76	mpbir2and 713	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐺 ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→𝐷)
78		simpl3 1194	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → 𝐻:𝐸–1-1-onto→𝐼)
79		simprr 773	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → 𝐷 ⊆ 𝐸)
80		f1ocoima 7321	. . 3 ⊢ (((𝐺 ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼 ∧ 𝐷 ⊆ 𝐸) → (𝐻 ∘ (𝐺 ∘ ^◡𝐹)):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))
81	77, 78, 79, 80	syl3anc 1373	. 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → (𝐻 ∘ (𝐺 ∘ ^◡𝐹)):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))
82		coass 6283	. . 3 ⊢ ((𝐻 ∘ 𝐺) ∘ ^◡𝐹) = (𝐻 ∘ (𝐺 ∘ ^◡𝐹))
83		f1oeq1 6834	. . 3 ⊢ (((𝐻 ∘ 𝐺) ∘ ^◡𝐹) = (𝐻 ∘ (𝐺 ∘ ^◡𝐹)) → (((𝐻 ∘ 𝐺) ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷) ↔ (𝐻 ∘ (𝐺 ∘ ^◡𝐹)):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷)))
84	82, 83	ax-mp 5	. 2 ⊢ (((𝐻 ∘ 𝐺) ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷) ↔ (𝐻 ∘ (𝐺 ∘ ^◡𝐹)):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))
85	81, 84	sylibr 234	1 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐻 ∘ 𝐺) ∘ ^◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∩ cin 3949 ⊆ wss 3950 ^◡ccnv 5682 dom cdm 5683 ran crn 5684 ↾ cres 5685 “ cima 5686 ∘ ccom 5687 Rel wrel 5688 Fn wfn 6554 ⟶wf 6555 –1-1→wf1 6556 –onto→wfo 6557 –1-1-onto→wf1o 6558

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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430

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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567

This theorem is referenced by: 3f1oss2 47061 uspgrlim 47932

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