Theorem actfunsnf1o 30682

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 3760	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3	2	cbvmptv 4750	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
4	1, 3	eqtri 2644	. 2 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5		vex 3203	. . . 4 ⊢ 𝑧 ∈ V
6		snex 4908	. . . 4 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
7	5, 6	unex 6956	. . 3 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
8	7	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9		vex 3203	. . . 4 ⊢ 𝑦 ∈ V
10	9	resex 5443	. . 3 ⊢ (𝑦 ↾ 𝐵) ∈ V
11	10	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ V)
12		rspe 3003	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
13	4, 7	elrnmpti 5376	. . . . . . 7 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
14	12, 13	sylibr 224	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
15	14	adantll 750	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16		simpr 477	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
17	16	reseq1d 5395	. . . . . 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 3603	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑𝑚 𝐵))
20		elmapfn 7880	. . . . . . . . 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 5938	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
24	22, 23	sylan 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
25	24	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
27		disjsn 4246	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
28	26, 27	sylibr 224	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
29	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵 ∩ {𝐼}) = ∅)
30	29	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31		fnunres1 29417	. . . . . . . 8 ⊢ ((𝑧 Fn 𝐵 ∧ {⟨𝐼, 𝑘⟩} Fn {𝐼} ∧ (𝐵 ∩ {𝐼}) = ∅) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
32	21, 25, 30, 31	syl3anc 1326	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
33	32	adantr 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
34	17, 33	eqtr2d 2657	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦 ↾ 𝐵))
35	15, 34	jca 554	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
36	35	anasss 679	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
37		simpr 477	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 = (𝑦 ↾ 𝐵))
38		simpr 477	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
39	38	reseq1d 5395	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
40	18	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶 ↑𝑚 𝐵))
41		simplr 792	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ 𝐴)
42	40, 41	sseldd 3604	. . . . . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼 ∈ 𝑉)
45		simp-4r 807	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑘 ∈ 𝐶)
46	44, 45, 23	syl2anc 693	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
47	28	ad4antr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝐵 ∩ {𝐼}) = ∅)
48	43, 46, 47, 31	syl3anc 1326	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
49	48, 41	eqeltrd 2701	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
50	39, 49	eqeltrd 2701	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) ∈ 𝐴)
51		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
52	51, 13	sylib 208	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
53	50, 52	r19.29a 3078	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ 𝐴)
54	53	adantr 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑦 ↾ 𝐵) ∈ 𝐴)
55	37, 54	eqeltrd 2701	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 ∈ 𝐴)
56	37	uneq1d 3766	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}))
57	39, 48	eqtrd 2656	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = 𝑧)
58	57	uneq1d 3766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
59	58, 38	eqtr4d 2659	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
60	59, 52	r19.29a 3078	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
61	60	adantr 481	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
62	56, 61	eqtr2d 2657	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
63	55, 62	jca 554	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
64	63	anasss 679	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
65	36, 64	impbida 877	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))))
66	4, 8, 11, 65	f1od 6885	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   ↦ cmpt 4729  ran crn 5115   ↾ cres 5116   Fn wfn 5883  –1-1-onto→wf1o 5887  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  breprexplema  30708