Theorem isf32lem9 10275

Description: Lemma for isfin3-2 10281. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
isf32lem.a	⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
isf32lem.b	⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
isf32lem.c	⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
isf32lem.d	⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}
isf32lem.e	⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢))
isf32lem.f	⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)
isf32lem.g	⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))))

Assertion

Ref	Expression
isf32lem9	⊢ (𝜑 → 𝐿:𝐺–onto→ω)

Distinct variable groups:   𝑥,𝑤   𝑡,𝐺   𝑥,𝐿   𝑡,𝑠,𝑢,𝑣,𝑤,𝑥,𝑦,𝜑   𝑤,𝐹,𝑥,𝑦   𝑆,𝑠,𝑡,𝑢,𝑣,𝑤,𝑥,𝑦   𝐽,𝑠,𝑡,𝑤,𝑥,𝑦   𝐾,𝑠,𝑡,𝑥,𝑦

Allowed substitution hints:   𝐹(𝑣,𝑢,𝑡,𝑠)   𝐺(𝑥,𝑦,𝑤,𝑣,𝑢,𝑠)   𝐽(𝑣,𝑢)   𝐾(𝑤,𝑣,𝑢)   𝐿(𝑦,𝑤,𝑣,𝑢,𝑡,𝑠)

Proof of Theorem isf32lem9

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isf32lem.g	. . . 4 ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))))
2		ssab2 4032	. . . . . . 7 ⊢ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))} ⊆ ω
3		iotacl 6479	. . . . . . 7 ⊢ (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))})
4	2, 3	sselid 3932	. . . . . 6 ⊢ (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ ω)
5		iotanul 6473	. . . . . . 7 ⊢ (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) = ∅)
6		peano1 7833	. . . . . . 7 ⊢ ∅ ∈ ω
7	5, 6	eqeltrdi 2845	. . . . . 6 ⊢ (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ ω)
8	4, 7	pm2.61i 182	. . . . 5 ⊢ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ ω
9	8	a1i 11	. . . 4 ⊢ (𝑡 ∈ 𝐺 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ ω)
10	1, 9	fmpti 7059	. . 3 ⊢ 𝐿:𝐺⟶ω
11	10	a1i 11	. 2 ⊢ (𝜑 → 𝐿:𝐺⟶ω)
12		isf32lem.a	. . . . . 6 ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
13		isf32lem.b	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
14		isf32lem.c	. . . . . 6 ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)
15		isf32lem.d	. . . . . 6 ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)}
16		isf32lem.e	. . . . . 6 ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢))
17		isf32lem.f	. . . . . 6 ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)
18	12, 13, 14, 15, 16, 17	isf32lem6 10272	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝐾‘𝑎) ≠ ∅)
19		n0 4306	. . . . 5 ⊢ ((𝐾‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐾‘𝑎))
20	18, 19	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 𝑏 ∈ (𝐾‘𝑎))
21	12, 13, 14, 15, 16, 17	isf32lem8 10274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝐾‘𝑎) ⊆ 𝐺)
22	21	sselda 3934	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → 𝑏 ∈ 𝐺)
23		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 ∈ (𝐾‘𝑠) ↔ 𝑏 ∈ (𝐾‘𝑠)))
24	23	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → ((𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) ↔ (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
25	24	iotabidv 6477	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑏 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
26		iotaex 6469	. . . . . . . . . . 11 ⊢ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ V
27	25, 1, 26	fvmpt3i 6948	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐺 → (𝐿‘𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
28	22, 27	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → (𝐿‘𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
29		simp1r 1200	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → 𝑏 ∈ (𝐾‘𝑎))
30		simpl1 1193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝜑)
31		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑠 ≠ 𝑎)
32	31	necomd 2988	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑎 ≠ 𝑠)
33		simpl2 1194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑎 ∈ ω)
34		simpl3 1195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑠 ∈ ω)
35	12, 13, 14, 15, 16, 17	isf32lem7 10273	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ≠ 𝑠) ∧ (𝑎 ∈ ω ∧ 𝑠 ∈ ω)) → ((𝐾‘𝑎) ∩ (𝐾‘𝑠)) = ∅)
36	30, 32, 33, 34, 35	syl22anc 839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → ((𝐾‘𝑎) ∩ (𝐾‘𝑠)) = ∅)
37		disj1 4405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐾‘𝑎) ∩ (𝐾‘𝑠)) = ∅ ↔ ∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠)))
38	36, 37	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → ∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠)))
39	38	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠 ≠ 𝑎 → ∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠))))
40		sp 2191	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠)) → (𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠)))
41	39, 40	syl6 35	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠 ≠ 𝑎 → (𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠))))
42	41	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾‘𝑎) → (𝑠 ≠ 𝑎 → ¬ 𝑏 ∈ (𝐾‘𝑠))))
43	42	3adant1r 1179	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾‘𝑎) → (𝑠 ≠ 𝑎 → ¬ 𝑏 ∈ (𝐾‘𝑠))))
44	29, 43	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠 ≠ 𝑎 → ¬ 𝑏 ∈ (𝐾‘𝑠)))
45	44	necon4ad 2952	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾‘𝑠) → 𝑠 = 𝑎))
46	45	3expia 1122	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 ∈ ω → (𝑏 ∈ (𝐾‘𝑠) → 𝑠 = 𝑎)))
47	46	impd 410	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠)) → 𝑠 = 𝑎))
48		eleq1w 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → (𝑠 ∈ ω ↔ 𝑎 ∈ ω))
49		fveq2 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑎 → (𝐾‘𝑠) = (𝐾‘𝑎))
50	49	eleq2d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → (𝑏 ∈ (𝐾‘𝑠) ↔ 𝑏 ∈ (𝐾‘𝑎)))
51	48, 50	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑎 → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠)) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑎))))
52	51	biimprcd 250	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑎)) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
53	52	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (𝐾‘𝑎) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
54	53	adantll 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
55	47, 54	impbid 212	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠)) ↔ 𝑠 = 𝑎))
56	55	iota5 6476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))) = 𝑎)
57	56	an32s 653	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))) = 𝑎)
58	28, 57	eqtr2d 2773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → 𝑎 = (𝐿‘𝑏))
59	22, 58	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → (𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏)))
60	59	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝑏 ∈ (𝐾‘𝑎) → (𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏))))
61	60	eximdv 1919	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾‘𝑎) → ∃𝑏(𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏))))
62		df-rex 3062	. . . . 5 ⊢ (∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏) ↔ ∃𝑏(𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏)))
63	61, 62	imbitrrdi 252	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾‘𝑎) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏)))
64	20, 63	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏))
65	64	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ω ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏))
66		dffo3 7049	. 2 ⊢ (𝐿:𝐺–onto→ω ↔ (𝐿:𝐺⟶ω ∧ ∀𝑎 ∈ ω ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏)))
67	11, 65, 66	sylanbrc 584	1 ⊢ (𝜑 → 𝐿:𝐺–onto→ω)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4286  𝒫 cpw 4555  ∩ cint 4903   class class class wbr 5099   ↦ cmpt 5180  ran crn 5626   ∘ ccom 5629  suc csuc 6320  ℩cio 6447  ⟶wf 6489  –onto→wfo 6491  ‘cfv 6493  ℩crio 7316  ωcom 7810   ≈ cen 8884

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9855

This theorem is referenced by:  isf32lem10  10276