Theorem isf32lem9 9864

Description: Lemma for isfin3-2 9870. 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 3969	. . . . . . 7 ⊢ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))} ⊆ ω
3		iotacl 6326	. . . . . . 7 ⊢ (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))})
4	2, 3	sseldi 3876	. . . . . 6 ⊢ (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ ω)
5		iotanul 6318	. . . . . . 7 ⊢ (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) = ∅)
6		peano1 7623	. . . . . . 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 185	. . . . 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 6889	. . 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 9861	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝐾‘𝑎) ≠ ∅)
19		n0 4236	. . . . 5 ⊢ ((𝐾‘𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐾‘𝑎))
20	18, 19	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → ∃𝑏 𝑏 ∈ (𝐾‘𝑎))
21	12, 13, 14, 15, 16, 17	isf32lem8 9863	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝐾‘𝑎) ⊆ 𝐺)
22	21	sselda 3878	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → 𝑏 ∈ 𝐺)
23		eleq1w 2816	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑏 → (𝑡 ∈ (𝐾‘𝑠) ↔ 𝑏 ∈ (𝐾‘𝑠)))
24	23	anbi2d 632	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑏 → ((𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)) ↔ (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
25	24	iotabidv 6324	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑏 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
26		iotaex 6320	. . . . . . . . . . 11 ⊢ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠))) ∈ V
27	25, 1, 26	fvmpt3i 6783	. . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → 𝑠 ≠ 𝑎)
32	31	necomd 2990	. . . . . . . . . . . . . . . . . . . . . 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 9862	. . . . . . . . . . . . . . . . . . . . . 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 4341	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐾‘𝑎) ∩ (𝐾‘𝑠)) = ∅ ↔ ∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠)))
38	36, 37	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠 ≠ 𝑎) → ∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠)))
39	38	ex 416	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠 ≠ 𝑎 → ∀𝑏(𝑏 ∈ (𝐾‘𝑎) → ¬ 𝑏 ∈ (𝐾‘𝑠))))
40		sp 2184	. . . . . . . . . . . . . . . . . . 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 2954	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠)) → 𝑠 = 𝑎))
48		eleq1w 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → (𝑠 ∈ ω ↔ 𝑎 ∈ ω))
49		fveq2 6677	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑎 → (𝐾‘𝑠) = (𝐾‘𝑎))
50	49	eleq2d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑎 → (𝑏 ∈ (𝐾‘𝑠) ↔ 𝑏 ∈ (𝐾‘𝑎)))
51	48, 50	anbi12d 634	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑎 → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠)) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑎))))
52	51	biimprcd 253	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑎)) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠))))
53	52	ancoms 462	. . . . . . . . . . . . 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 215	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐾‘𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾‘𝑠)) ↔ 𝑠 = 𝑎))
56	55	iota5 6323	. . . . . . . . . 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 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → 𝑎 = (𝐿‘𝑏))
59	22, 58	jca 515	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾‘𝑎)) → (𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏)))
60	59	ex 416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (𝑏 ∈ (𝐾‘𝑎) → (𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏))))
61	60	eximdv 1924	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾‘𝑎) → ∃𝑏(𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏))))
62		df-rex 3060	. . . . 5 ⊢ (∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏) ↔ ∃𝑏(𝑏 ∈ 𝐺 ∧ 𝑎 = (𝐿‘𝑏)))
63	61, 62	syl6ibr 255	. . . 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 3097	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ω ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏))
66		dffo3 6881	. 2 ⊢ (𝐿:𝐺–onto→ω ↔ (𝐿:𝐺⟶ω ∧ ∀𝑎 ∈ ω ∃𝑏 ∈ 𝐺 𝑎 = (𝐿‘𝑏)))
67	11, 65, 66	sylanbrc 586	1 ⊢ (𝜑 → 𝐿:𝐺–onto→ω)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1088  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2114  ∃!weu 2570  {cab 2717   ≠ wne 2935  ∀wral 3054  ∃wrex 3055  {crab 3058   ∖ cdif 3841   ∩ cin 3843   ⊆ wss 3844   ⊊ wpss 3845  ∅c0 4212  𝒫 cpw 4489  ∩ cint 4837   class class class wbr 5031   ↦ cmpt 5111  ran crn 5527   ∘ ccom 5530  suc csuc 6175  ℩cio 6296  ⟶wf 6336  –onto→wfo 6338  ‘cfv 6340  ℩crio 7129  ωcom 7602   ≈ cen 8555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-se 5485  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7130  df-om 7603  df-wrecs 7979  df-recs 8040  df-1o 8134  df-er 8323  df-en 8559  df-dom 8560  df-sdom 8561  df-fin 8562  df-card 9444

This theorem is referenced by:  isf32lem10  9865