Theorem actfunsnf1o 31985

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 4083	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3	2	cbvmptv 5133	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
4	1, 3	eqtri 2821	. 2 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5		vex 3444	. . . 4 ⊢ 𝑧 ∈ V
6		snex 5297	. . . 4 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
7	5, 6	unex 7449	. . 3 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
8	7	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9		vex 3444	. . . 4 ⊢ 𝑦 ∈ V
10	9	resex 5866	. . 3 ⊢ (𝑦 ↾ 𝐵) ∈ V
11	10	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ V)
12		rspe 3263	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
13	4, 7	elrnmpti 5796	. . . . . . 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 713	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16		simpr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
17	16	reseq1d 5817	. . . . . 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 3915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
20		elmapfn 8412	. . . . . . . . 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 6376	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
24	22, 23	sylan 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
25	24	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
27		disjsn 4607	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
28	26, 27	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
29	28	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵 ∩ {𝐼}) = ∅)
30	29	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31		fnunres1 30369	. . . . . . . 8 ⊢ ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
32	21, 25, 30, 31	syl3anc 1368	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
33	32	adantr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
34	17, 33	eqtr2d 2834	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦 ↾ 𝐵))
35	15, 34	jca 515	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
36	35	anasss 470	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
37		simpr 488	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 = (𝑦 ↾ 𝐵))
38		simpr 488	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
39	38	reseq1d 5817	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
40	18	ad3antrrr 729	. . . . . . . . . . . . 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 3916	. . . . . . . . . . . 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 731	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
47	28	ad4antr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
48	43, 46, 47, 31	syl3anc 1368	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
49	48, 41	eqeltrd 2890	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
50	39, 49	eqeltrd 2890	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) ∈ 𝐴)
51		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
52	51, 13	sylib 221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
53	50, 52	r19.29a 3248	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ 𝐴)
54	53	adantr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑦 ↾ 𝐵) ∈ 𝐴)
55	37, 54	eqeltrd 2890	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 ∈ 𝐴)
56	37	uneq1d 4089	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}))
57	39, 48	eqtrd 2833	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = 𝑧)
58	57	uneq1d 4089	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
59	58, 38	eqtr4d 2836	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
60	59, 52	r19.29a 3248	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
61	60	adantr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
62	56, 61	eqtr2d 2834	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
63	55, 62	jca 515	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
64	63	anasss 470	. . 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 7377	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521   Fn wfn 6319  –1-1-onto→wf1o 6323  (class class class)co 7135   ↑_m cmap 8389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391

This theorem is referenced by:  breprexplema  32011