Theorem actfunsnf1o 34636

Description: The action 𝐹 of extending function from 𝐵 to 𝐶 with new values at point 𝐼 is a bijection. (Contributed by Thierry Arnoux, 9-Dec-2021.)

Hypotheses

Ref	Expression
actfunsn.1	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
actfunsn.2	⊢ (𝜑 → 𝐶 ∈ V)
actfunsn.3	⊢ (𝜑 → 𝐼 ∈ 𝑉)
actfunsn.4	⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
actfunsn.5	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))

Assertion

Ref	Expression
actfunsnf1o	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑘,𝐼,𝑥   𝜑,𝑘

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑘)   𝐵(𝑥,𝑘)   𝐶(𝑥,𝑘)   𝐹(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem actfunsnf1o

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		actfunsn.5	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))
2		uneq1 4136	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3	2	cbvmptv 5225	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
4	1, 3	eqtri 2758	. 2 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5		vex 3463	. . . 4 ⊢ 𝑧 ∈ V
6		snex 5406	. . . 4 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
7	5, 6	unex 7738	. . 3 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
8	7	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9		vex 3463	. . . 4 ⊢ 𝑦 ∈ V
10	9	resex 6016	. . 3 ⊢ (𝑦 ↾ 𝐵) ∈ V
11	10	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ V)
12		rspe 3232	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
13	4, 7	elrnmpti 5942	. . . . . . 7 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
14	12, 13	sylibr 234	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
15	14	adantll 714	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
17	16	reseq1d 5965	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
18		actfunsn.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
19	18	sselda 3958	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
20		elmapfn 8879	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐶 ↑m 𝐵) → 𝑧 Fn 𝐵)
21	19, 20	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 Fn 𝐵)
22		actfunsn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
23		fnsng 6588	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
24	22, 23	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
25	24	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
27		disjsn 4687	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
28	26, 27	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵 ∩ {𝐼}) = ∅)
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31		fnunres1 6650	. . . . . . . 8 ⊢ ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
32	21, 25, 30, 31	syl3anc 1373	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
33	32	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
34	17, 33	eqtr2d 2771	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦 ↾ 𝐵))
35	15, 34	jca 511	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
36	35	anasss 466	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
37		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 = (𝑦 ↾ 𝐵))
38		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
39	38	reseq1d 5965	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
40	18	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
41		simplr 768	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ 𝐴)
42	40, 41	sseldd 3959	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ (𝐶 ↑m 𝐵))
43	42, 20	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 Fn 𝐵)
44	22	ad4antr 732	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼 ∈ 𝑉)
45		simp-4r 783	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑘 ∈ 𝐶)
46	44, 45, 23	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
47	28	ad4antr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
48	43, 46, 47, 31	syl3anc 1373	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
49	48, 41	eqeltrd 2834	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
50	39, 49	eqeltrd 2834	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) ∈ 𝐴)
51		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
52	51, 13	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
53	50, 52	r19.29a 3148	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ 𝐴)
54	53	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑦 ↾ 𝐵) ∈ 𝐴)
55	37, 54	eqeltrd 2834	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 ∈ 𝐴)
56	37	uneq1d 4142	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}))
57	39, 48	eqtrd 2770	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = 𝑧)
58	57	uneq1d 4142	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
59	58, 38	eqtr4d 2773	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
60	59, 52	r19.29a 3148	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
61	60	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
62	56, 61	eqtr2d 2771	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
63	55, 62	jca 511	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
64	63	anasss 466	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
65	36, 64	impbida 800	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))))
66	4, 8, 11, 65	f1od 7659	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  {csn 4601  ⟨cop 4607   ↦ cmpt 5201  ran crn 5655   ↾ cres 5656   Fn wfn 6526  –1-1-onto→wf1o 6530  (class class class)co 7405   ↑_m cmap 8840

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 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-map 8842

This theorem is referenced by:  breprexplema  34662