Theorem actfunsnf1o 35233

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 4160	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3	2	cbvmptv 5208	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
4	1, 3	eqtri 2783	. 2 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5		vex 3454	. . . 4 ⊢ 𝑧 ∈ V
6		snex 5396	. . . 4 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
7	5, 6	unex 7761	. . 3 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
8	7	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
10	9	resex 6017	. . 3 ⊢ (𝑦 ↾ 𝐵) ∈ V
11	10	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ V)
12		rspe 3252	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
13	4, 7	elrnmpti 5943	. . . . . . 7 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
14	12, 13	sylibr 237	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
15	14	adantll 727	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
17	16	reseq1d 5968	. . . . . 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 3908	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
20		elmapfn 8887	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐶 ↑m 𝐵) → 𝑧 Fn 𝐵)
21	19, 20	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 Fn 𝐵)
22		actfunsn.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
23		fnsng 6592	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
24	22, 23	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
25	24	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
27		disjsn 4671	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
28	26, 27	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵 ∩ {𝐼}) = ∅)
30	29	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31		fnunres1 6651	. . . . . . . 8 ⊢ ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
32	21, 25, 30, 31	syl3anc 1398	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
33	32	adantr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
34	17, 33	eqtr2d 2796	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦 ↾ 𝐵))
35	15, 34	jca 521	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
36	35	anasss 472	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
37		simpr 490	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 = (𝑦 ↾ 𝐵))
38		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
39	38	reseq1d 5968	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
40	18	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
41		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ 𝐴)
42	40, 41	sseldd 3909	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ (𝐶 ↑m 𝐵))
43	42, 20	syl 18	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 Fn 𝐵)
44	22	ad4antr 745	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼 ∈ 𝑉)
45		simp-4r 796	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑘 ∈ 𝐶)
46	44, 45, 23	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
47	28	ad4antr 745	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
48	43, 46, 47, 31	syl3anc 1398	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
49	48, 41	eqeltrd 2860	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
50	39, 49	eqeltrd 2860	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) ∈ 𝐴)
51	13	bilani 510	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
52	50, 51	r19.29a 3170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ 𝐴)
53	52	adantr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑦 ↾ 𝐵) ∈ 𝐴)
54	37, 53	eqeltrd 2860	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 ∈ 𝐴)
55	37	uneq1d 4166	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}))
56	39, 48	eqtrd 2795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = 𝑧)
57	56	uneq1d 4166	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
58	57, 38	eqtr4d 2798	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
59	58, 51	r19.29a 3170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
60	59	adantr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
61	55, 60	eqtr2d 2796	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
62	54, 61	jca 521	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
63	62	anasss 472	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
64	36, 63	impbida 813	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))))
65	4, 8, 11, 64	f1od 7673	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589   ↦ cmpt 5185  ran crn 5651   ↾ cres 5652   Fn wfn 6533  –1-1-onto→wf1o 6537  (class class class)co 7420   ↑_m cmap 8847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849

This theorem is used by:  breprexplema  35259