Theorem isf32lem9 10269

Description: Lemma for isfin3-2 10275. 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 4029	. . . . . . 7 ⊢ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))} ⊆ ω
3		iotacl 6476	. . . . . . 7 ⊢ (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))})
4	2, 3	sselid 3929	. . . . . 6 ⊢ (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ ω)
5		iotanul 6470	. . . . . . 7 ⊢ (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) = ∅)
6		peano1 7829	. . . . . . 7 ⊢ ∅ ∈ ω
7	5, 6	eqeltrdi 2842	. . . . . 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 7055	. . 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 10266	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝐾‘𝑎) ≠ ∅)
19		n0 4303	. . . . 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 10268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝐾‘𝑎) ⊆ 𝐺)
22	21	sselda 3931	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → 𝑏 ∈ 𝐺)
23		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 ∈ (𝐾‘𝑠) ↔ 𝑏 ∈ (𝐾‘𝑠)))
24	23	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → ((𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) ↔ (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
25	24	iotabidv 6474	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑏 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
26		iotaex 6466	. . . . . . . . . . 11 ⊢ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ V
27	25, 1, 26	fvmpt3i 6944	. . . . . . . . . 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 1199	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → 𝑏 ∈ (𝐾‘𝑎))
30		simpl1 1192	. . . . . . . . . . . . . . . . . . . . . 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 2985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑎 ≠ 𝑠)
33		simpl2 1193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑎 ∈ ω)
34		simpl3 1194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑠 ∈ ω)
35	12, 13, 14, 15, 16, 17	isf32lem7 10267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ≠ 𝑠) ∧ (𝑎 ∈ ω ∧ 𝑠 ∈ ω)) → ((𝐾‘𝑎) ∩ (𝐾‘𝑠)) = ∅)
36	30, 32, 33, 34, 35	syl22anc 838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → ((𝐾‘𝑎) ∩ (𝐾‘𝑠)) = ∅)
37		disj1 4402	. . . . . . . . . . . . . . . . . . . . 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 2188	. . . . . . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . . . . 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 2949	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾‘𝑠) → 𝑠 = 𝑎))
46	45	3expia 1121	. . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → (𝑠 ∈ ω ↔ 𝑎 ∈ ω))
49		fveq2 6832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑎 → (𝐾‘𝑠) = (𝐾‘𝑎))
50	49	eleq2d 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → (𝑏 ∈ (𝐾‘𝑠) ↔ 𝑏 ∈ (𝐾‘𝑎)))
51	48, 50	anbi12d 632	. . . . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 6473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))) = 𝑎)
57	56	an32s 652	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))) = 𝑎)
58	28, 57	eqtr2d 2770	. . . . . . . 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 1918	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾‘𝑎) → ∃𝑏(𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏))))
62		df-rex 3059	. . . . 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 3126	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ω ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏))
66		dffo3 7045	. 2 ⊢ (𝐿:𝐺–onto→ω ↔ (𝐿:𝐺⟶ω ∧ ∀𝑎 ∈ ω ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏)))
67	11, 65, 66	sylanbrc 583	1 ⊢ (𝜑 → 𝐿:𝐺–onto→ω)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2566  {cab 2712   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  {crab 3397   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899   ⊊ wpss 3900  ∅c0 4283  𝒫 cpw 4552  ∩ cint 4900   class class class wbr 5096   ↦ cmpt 5177  ran crn 5623   ∘ ccom 5626  suc csuc 6317  ℩cio 6444  ⟶wf 6486  –onto→wfo 6488  ‘cfv 6490  ℩crio 7312  ωcom 7806   ≈ cen 8878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849

This theorem is referenced by:  isf32lem10  10270