Theorem actfunsnf1o 34241

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 4155	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
3	2	cbvmptv 5263	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
4	1, 3	eqtri 2755	. 2 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
5		vex 3475	. . . 4 ⊢ 𝑧 ∈ V
6		snex 5435	. . . 4 ⊢ {⟨𝐼, 𝑘⟩} ∈ V
7	5, 6	unex 7752	. . 3 ⊢ (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V
8	7	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) ∈ V)
9		vex 3475	. . . 4 ⊢ 𝑦 ∈ V
10	9	resex 6036	. . 3 ⊢ (𝑦 ↾ 𝐵) ∈ V
11	10	a1i 11	. 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ V)
12		rspe 3242	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
13	4, 7	elrnmpti 5964	. . . . . . 7 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
14	12, 13	sylibr 233	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
15	14	adantll 712	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 ∈ ran 𝐹)
16		simpr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
17	16	reseq1d 5986	. . . . . 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 3980	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ (𝐶 ↑m 𝐵))
20		elmapfn 8888	. . . . . . . . 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 6608	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
24	22, 23	sylan 578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
25	24	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
26		actfunsn.4	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵)
27		disjsn 4718	. . . . . . . . . . 11 ⊢ ((𝐵 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ 𝐵)
28	26, 27	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ {𝐼}) = ∅)
29	28	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → (𝐵 ∩ {𝐼}) = ∅)
30	29	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) → (𝐵 ∩ {𝐼}) = ∅)
31		fnunres1 6669	. . . . . . . 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 479	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) = 𝑧)
34	17, 33	eqtr2d 2768	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 = (𝑦 ↾ 𝐵))
35	15, 34	jca 510	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
36	35	anasss 465	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))) → (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵)))
37		simpr 483	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 = (𝑦 ↾ 𝐵))
38		simpr 483	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
39	38	reseq1d 5986	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵))
40	18	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐴 ⊆ (𝐶 ↑m 𝐵))
41		simplr 767	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑧 ∈ 𝐴)
42	40, 41	sseldd 3981	. . . . . . . . . . . 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 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝐼 ∈ 𝑉)
45		simp-4r 782	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → 𝑘 ∈ 𝐶)
46	44, 45, 23	syl2anc 582	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → {⟨𝐼, 𝑘⟩} Fn {𝐼})
47	28	ad4antr 730	. . . . . . . . . . 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 2828	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑧 ∪ {⟨𝐼, 𝑘⟩}) ↾ 𝐵) ∈ 𝐴)
50	39, 49	eqeltrd 2828	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) ∈ 𝐴)
51		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ran 𝐹)
52	51, 13	sylib 217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
53	50, 52	r19.29a 3158	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → (𝑦 ↾ 𝐵) ∈ 𝐴)
54	53	adantr 479	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑦 ↾ 𝐵) ∈ 𝐴)
55	37, 54	eqeltrd 2828	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑧 ∈ 𝐴)
56	37	uneq1d 4161	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∪ {⟨𝐼, 𝑘⟩}) = ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}))
57	39, 48	eqtrd 2767	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → (𝑦 ↾ 𝐵) = 𝑧)
58	57	uneq1d 4161	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
59	58, 38	eqtr4d 2770	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
60	59, 52	r19.29a 3158	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
61	60	adantr 479	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → ((𝑦 ↾ 𝐵) ∪ {⟨𝐼, 𝑘⟩}) = 𝑦)
62	56, 61	eqtr2d 2768	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩}))
63	55, 62	jca 510	. . . 4 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 = (𝑦 ↾ 𝐵)) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
64	63	anasss 465	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐶) ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))) → (𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})))
65	36, 64	impbida 799	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → ((𝑧 ∈ 𝐴 ∧ 𝑦 = (𝑧 ∪ {⟨𝐼, 𝑘⟩})) ↔ (𝑦 ∈ ran 𝐹 ∧ 𝑧 = (𝑦 ↾ 𝐵))))
66	4, 8, 11, 65	f1od 7677	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3066  Vcvv 3471   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4324  {csn 4630  ⟨cop 4636   ↦ cmpt 5233  ran crn 5681   ↾ cres 5682   Fn wfn 6546  –1-1-onto→wf1o 6550  (class class class)co 7424   ↑_m cmap 8849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-map 8851

This theorem is referenced by:  breprexplema  34267