domss2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > domss2		Structured version Visualization version GIF version

Theorem domss2 8407

Description: A corollary of disjenex 8406. If 𝐹 is an injection from 𝐴 to 𝐵 then 𝐺 is a right inverse of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

**Hypothesis**
Ref	Expression
domss2.1	⊢ 𝐺 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))

**Assertion**
Ref	Expression
domss2	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran 𝐺 ∧ 𝐴 ⊆ ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)))

**Proof of Theorem domss2**
Step	Hyp	Ref	Expression
1		f1f1orn 6402	. . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
2	1	3ad2ant1 1124	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→ran 𝐹)
3		simp2 1128	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
4		rnexg 7376	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V)
6		uniexg 7232	. . . . . . . . 9 ⊢ (ran 𝐴 ∈ V → ∪ ran 𝐴 ∈ V)
7		pwexg 5090	. . . . . . . . 9 ⊢ (∪ ran 𝐴 ∈ V → 𝒫 ∪ ran 𝐴 ∈ V)
8	5, 6, 7	3syl 18	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 ∪ ran 𝐴 ∈ V)
9		1stconst 7546	. . . . . . . 8 ⊢ (𝒫 ∪ ran 𝐴 ∈ V → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹))
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹))
11		difexg 5045	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
12	11	3ad2ant3 1126	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
13		disjen 8405	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∖ ran 𝐹) ∈ V) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹)))
14	3, 12, 13	syl2anc 579	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹)))
15	14	simpld 490	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅)
16		disjdif 4263	. . . . . . . 8 ⊢ (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅
17	16	a1i 11	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅)
18		f1oun 6410	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→ran 𝐹 ∧ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹)) ∧ ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅)) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)))
19	2, 10, 15, 17, 18	syl22anc 829	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)))
20		undif2 4267	. . . . . . . 8 ⊢ (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = (ran 𝐹 ∪ 𝐵)
21		f1f 6351	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
22	21	3ad2ant1 1124	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹:𝐴⟶𝐵)
23	22	frnd 6298	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐹 ⊆ 𝐵)
24		ssequn1 4005	. . . . . . . . 9 ⊢ (ran 𝐹 ⊆ 𝐵 ↔ (ran 𝐹 ∪ 𝐵) = 𝐵)
25	23, 24	sylib 210	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∪ 𝐵) = 𝐵)
26	20, 25	syl5eq 2825	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = 𝐵)
27	26	f1oeq3d 6388	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) ↔ (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵))
28	19, 27	mpbid 224	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵)
29		f1ocnv 6403	. . . . 5 ⊢ ((𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵 → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
30	28, 29	syl 17	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
31		domss2.1	. . . . 5 ⊢ 𝐺 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
32		f1oeq1 6380	. . . . 5 ⊢ (𝐺 = ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → (𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↔ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))))
33	31, 32	ax-mp 5	. . . 4 ⊢ (𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↔ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
34	30, 33	sylibr 226	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
35		f1ofo 6398	. . . . 5 ⊢ (𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) → 𝐺:𝐵–onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
36		forn 6369	. . . . 5 ⊢ (𝐺:𝐵–onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) → ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
37	34, 35, 36	3syl 18	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
38	37	f1oeq3d 6388	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran 𝐺 ↔ 𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))))
39	34, 38	mpbird 249	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐺:𝐵–1-1-onto→ran 𝐺)
40		ssun1 3998	. . 3 ⊢ 𝐴 ⊆ (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))
41	40, 37	syl5sseqr 3872	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ⊆ ran 𝐺)
42		ssid 3841	. . . 4 ⊢ ran 𝐹 ⊆ ran 𝐹
43		cores 5892	. . . 4 ⊢ (ran 𝐹 ⊆ ran 𝐹 → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹))
44	42, 43	ax-mp 5	. . 3 ⊢ ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹)
45		dmres 5668	. . . . . . . . 9 ⊢ dom (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = (ran 𝐹 ∩ dom ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
46		f1ocnv 6403	. . . . . . . . . . . 12 ⊢ ((1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹) → ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):(𝐵 ∖ ran 𝐹)–1-1-onto→((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))
47		f1odm 6395	. . . . . . . . . . . 12 ⊢ (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):(𝐵 ∖ ran 𝐹)–1-1-onto→((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) → dom ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = (𝐵 ∖ ran 𝐹))
48	10, 46, 47	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = (𝐵 ∖ ran 𝐹))
49	48	ineq2d 4036	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∩ dom ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)))
50	49, 16	syl6eq 2829	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∩ dom ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = ∅)
51	45, 50	syl5eq 2825	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅)
52		relres 5675	. . . . . . . . 9 ⊢ Rel (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)
53		reldm0 5588	. . . . . . . . 9 ⊢ (Rel (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) → ((^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅ ↔ dom (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅))
54	52, 53	ax-mp 5	. . . . . . . 8 ⊢ ((^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅ ↔ dom (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅)
55	51, 54	sylibr 226	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅)
56	55	uneq2d 3989	. . . . . 6 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (^◡𝐹 ∪ (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) = (^◡𝐹 ∪ ∅))
57		cnvun 5792	. . . . . . . . 9 ⊢ ^◡(𝐹 ∪ (1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = (^◡𝐹 ∪ ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
58	31, 57	eqtri 2801	. . . . . . . 8 ⊢ 𝐺 = (^◡𝐹 ∪ ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))
59	58	reseq1i 5638	. . . . . . 7 ⊢ (𝐺 ↾ ran 𝐹) = ((^◡𝐹 ∪ ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ↾ ran 𝐹)
60		resundir 5661	. . . . . . 7 ⊢ ((^◡𝐹 ∪ ^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ↾ ran 𝐹) = ((^◡𝐹 ↾ ran 𝐹) ∪ (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹))
61		df-rn 5366	. . . . . . . . . 10 ⊢ ran 𝐹 = dom ^◡𝐹
62	61	reseq2i 5639	. . . . . . . . 9 ⊢ (^◡𝐹 ↾ ran 𝐹) = (^◡𝐹 ↾ dom ^◡𝐹)
63		relcnv 5757	. . . . . . . . . 10 ⊢ Rel ^◡𝐹
64		resdm 5691	. . . . . . . . . 10 ⊢ (Rel ^◡𝐹 → (^◡𝐹 ↾ dom ^◡𝐹) = ^◡𝐹)
65	63, 64	ax-mp 5	. . . . . . . . 9 ⊢ (^◡𝐹 ↾ dom ^◡𝐹) = ^◡𝐹
66	62, 65	eqtri 2801	. . . . . . . 8 ⊢ (^◡𝐹 ↾ ran 𝐹) = ^◡𝐹
67	66	uneq1i 3985	. . . . . . 7 ⊢ ((^◡𝐹 ↾ ran 𝐹) ∪ (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) = (^◡𝐹 ∪ (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹))
68	59, 60, 67	3eqtrri 2806	. . . . . 6 ⊢ (^◡𝐹 ∪ (^◡(1^st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) = (𝐺 ↾ ran 𝐹)
69		un0 4192	. . . . . 6 ⊢ (^◡𝐹 ∪ ∅) = ^◡𝐹
70	56, 68, 69	3eqtr3g 2836	. . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺 ↾ ran 𝐹) = ^◡𝐹)
71	70	coeq1d 5529	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (^◡𝐹 ∘ 𝐹))
72		f1cocnv1 6420	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
73	72	3ad2ant1 1124	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
74	71, 73	eqtrd 2813	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = ( I ↾ 𝐴))
75	44, 74	syl5eqr 2827	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
76	39, 41, 75	3jca 1119	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran 𝐺 ∧ 𝐴 ⊆ ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∖ cdif 3788 ∪ cun 3789 ∩ cin 3790 ⊆ wss 3791 ∅c0 4140 𝒫 cpw 4378 {csn 4397 ∪ cuni 4671 class class class wbr 4886 I cid 5260 × cxp 5353 ^◡ccnv 5354 dom cdm 5355 ran crn 5356 ↾ cres 5357 ∘ ccom 5359 Rel wrel 5360 ⟶wf 6131 –1-1→wf1 6132 –onto→wfo 6133 –1-1-onto→wf1o 6134 1^st c1st 7443 ≈ cen 8238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-1st 7445 df-2nd 7446 df-en 8242

This theorem is referenced by: domssex2 8408 domssex 8409

W3C validator