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Theorem axcc2lem 9258

Description: Lemma for axcc2 9259. (Contributed by Mario Carneiro, 8-Feb-2013.)

**Hypotheses**
Ref	Expression
axcc2lem.1	⊢ 𝐾 = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
axcc2lem.2	⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
axcc2lem.3	⊢ 𝐺 = (𝑛 ∈ ω ↦ (2^nd ‘(𝑓‘(𝐴‘𝑛))))

**Assertion**
Ref	Expression
axcc2lem	⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Distinct variable groups: 𝐴,𝑓,𝑛 𝑓,𝐹,𝑔 𝑔,𝐺,𝑛 𝑛,𝐾 Allowed substitution hints: 𝐴(𝑔) 𝐹(𝑛) 𝐺(𝑓) 𝐾(𝑓,𝑔)

Proof of Theorem axcc2lem Dummy variables 𝑎 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 ⊢ (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ V
2		axcc2lem.3	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ω ↦ (2^nd ‘(𝑓‘(𝐴‘𝑛))))
3	1, 2	fnmpti 6022	. . 3 ⊢ 𝐺 Fn ω
4		snex 4908	. . . . . . . . . . . . . . 15 ⊢ {𝑛} ∈ V
5		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐾‘𝑛) ∈ V
6	4, 5	xpex 6962	. . . . . . . . . . . . . 14 ⊢ ({𝑛} × (𝐾‘𝑛)) ∈ V
7		axcc2lem.2	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
8	7	fvmpt2 6291	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾‘𝑛)) ∈ V) → (𝐴‘𝑛) = ({𝑛} × (𝐾‘𝑛)))
9	6, 8	mpan2 707	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → (𝐴‘𝑛) = ({𝑛} × (𝐾‘𝑛)))
10		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑛 ∈ V
11	10	snnz 4309	. . . . . . . . . . . . . 14 ⊢ {𝑛} ≠ ∅
12		0ex 4790	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
13	12	snnz 4309	. . . . . . . . . . . . . . . . 17 ⊢ {∅} ≠ ∅
14		iftrue 4092	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = {∅})
15	14	neeq1d 2853	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) = ∅ → (if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
16	13, 15	mpbiri 248	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅)
17		iffalse 4095	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = (𝐹‘𝑛))
18		df-ne 2795	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑛) ≠ ∅ ↔ ¬ (𝐹‘𝑛) = ∅)
19	18	biimpri 218	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑛) = ∅ → (𝐹‘𝑛) ≠ ∅)
20	17, 19	eqnetrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅)
21	16, 20	pm2.61i 176	. . . . . . . . . . . . . . 15 ⊢ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅
22		p0ex 4853	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
23		fvex 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑛) ∈ V
24	22, 23	ifex 4156	. . . . . . . . . . . . . . . . 17 ⊢ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ∈ V
25		axcc2lem.1	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐾 = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
26	25	fvmpt2 6291	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ∈ V) → (𝐾‘𝑛) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
27	24, 26	mpan2 707	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → (𝐾‘𝑛) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
28	27	neeq1d 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → ((𝐾‘𝑛) ≠ ∅ ↔ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅))
29	21, 28	mpbiri 248	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → (𝐾‘𝑛) ≠ ∅)
30		xpnz 5553	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ≠ ∅ ∧ (𝐾‘𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾‘𝑛)) ≠ ∅)
31	30	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (({𝑛} ≠ ∅ ∧ (𝐾‘𝑛) ≠ ∅) → ({𝑛} × (𝐾‘𝑛)) ≠ ∅)
32	11, 29, 31	sylancr 695	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → ({𝑛} × (𝐾‘𝑛)) ≠ ∅)
33	9, 32	eqnetrd 2861	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → (𝐴‘𝑛) ≠ ∅)
34	6, 7	fnmpti 6022	. . . . . . . . . . . . . 14 ⊢ 𝐴 Fn ω
35		fnfvelrn 6356	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴‘𝑛) ∈ ran 𝐴)
36	34, 35	mpan 706	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → (𝐴‘𝑛) ∈ ran 𝐴)
37		neeq1 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑛) → (𝑧 ≠ ∅ ↔ (𝐴‘𝑛) ≠ ∅))
38		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑛) → (𝑓‘𝑧) = (𝑓‘(𝐴‘𝑛)))
39		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑛) → 𝑧 = (𝐴‘𝑛))
40	38, 39	eleq12d 2695	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑛) → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛)))
41	37, 40	imbi12d 334	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐴‘𝑛) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ((𝐴‘𝑛) ≠ ∅ → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))))
42	41	rspccv 3306	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝐴‘𝑛) ∈ ran 𝐴 → ((𝐴‘𝑛) ≠ ∅ → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))))
43	36, 42	syl5 34	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐴‘𝑛) ≠ ∅ → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))))
44	33, 43	mpdi 45	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑛 ∈ ω → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛)))
45	44	impcom 446	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))
46	9	eleq2d 2687	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛) ↔ (𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛))))
47	46	adantr 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛) ↔ (𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛))))
48	45, 47	mpbid 222	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛)))
49		xp2nd 7199	. . . . . . . . 9 ⊢ ((𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛)) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ (𝐾‘𝑛))
50	48, 49	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ (𝐾‘𝑛))
51	50	3adant3 1081	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ (𝐾‘𝑛))
52	2	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ V) → (𝐺‘𝑛) = (2^nd ‘(𝑓‘(𝐴‘𝑛))))
53	1, 52	mpan2 707	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (𝐺‘𝑛) = (2^nd ‘(𝑓‘(𝐴‘𝑛))))
54	53	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐺‘𝑛) = (2^nd ‘(𝑓‘(𝐴‘𝑛))))
55	54	eqcomd 2628	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) = (𝐺‘𝑛))
56	27	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐾‘𝑛) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
57		ifnefalse 4098	. . . . . . . . 9 ⊢ ((𝐹‘𝑛) ≠ ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = (𝐹‘𝑛))
58	57	3ad2ant3 1084	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = (𝐹‘𝑛))
59	56, 58	eqtrd 2656	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐾‘𝑛) = (𝐹‘𝑛))
60	51, 55, 59	3eltr3d 2715	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐺‘𝑛) ∈ (𝐹‘𝑛))
61	60	3expia 1267	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛)))
62	61	expcom 451	. . . 4 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))))
63	62	ralrimiv 2965	. . 3 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛)))
64		omex 8540	. . . . 5 ⊢ ω ∈ V
65		fnex 6481	. . . . 5 ⊢ ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
66	3, 64, 65	mp2an 708	. . . 4 ⊢ 𝐺 ∈ V
67		fneq1 5979	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
68		fveq1 6190	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑛) = (𝐺‘𝑛))
69	68	eleq1d 2686	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑛) ∈ (𝐹‘𝑛) ↔ (𝐺‘𝑛) ∈ (𝐹‘𝑛)))
70	69	imbi2d 330	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ↔ ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))))
71	70	ralbidv 2986	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))))
72	67, 71	anbi12d 747	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛))) ↔ (𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛)))))
73	66, 72	spcev 3300	. . 3 ⊢ ((𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛))))
74	3, 63, 73	sylancr 695	. 2 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛))))
75	6	a1i 11	. . . . . 6 ⊢ ((ω ∈ V ∧ 𝑛 ∈ ω) → ({𝑛} × (𝐾‘𝑛)) ∈ V)
76	75, 7	fmptd 6385	. . . . 5 ⊢ (ω ∈ V → 𝐴:ω⟶V)
77	64, 76	ax-mp 5	. . . 4 ⊢ 𝐴:ω⟶V
78		sneq 4187	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘})
79		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝐾‘𝑛) = (𝐾‘𝑘))
80	78, 79	xpeq12d 5140	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)))
81	80, 7, 6	fvmpt3i 6287	. . . . . . . 8 ⊢ (𝑘 ∈ ω → (𝐴‘𝑘) = ({𝑘} × (𝐾‘𝑘)))
82	81	adantl 482	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴‘𝑘) = ({𝑘} × (𝐾‘𝑘)))
83	82	eqeq2d 2632	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = (𝐴‘𝑘) ↔ (𝐴‘𝑛) = ({𝑘} × (𝐾‘𝑘))))
84	9	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴‘𝑛) = ({𝑛} × (𝐾‘𝑛)))
85	84	eqeq1d 2624	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = ({𝑘} × (𝐾‘𝑘)) ↔ ({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘))))
86		xp11 5569	. . . . . . . . . 10 ⊢ (({𝑛} ≠ ∅ ∧ (𝐾‘𝑛) ≠ ∅) → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾‘𝑛) = (𝐾‘𝑘))))
87	11, 29, 86	sylancr 695	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾‘𝑛) = (𝐾‘𝑘))))
88	10	sneqr 4371	. . . . . . . . . 10 ⊢ ({𝑛} = {𝑘} → 𝑛 = 𝑘)
89	88	adantr 481	. . . . . . . . 9 ⊢ (({𝑛} = {𝑘} ∧ (𝐾‘𝑛) = (𝐾‘𝑘)) → 𝑛 = 𝑘)
90	87, 89	syl6bi 243	. . . . . . . 8 ⊢ (𝑛 ∈ ω → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) → 𝑛 = 𝑘))
91	90	adantr 481	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) → 𝑛 = 𝑘))
92	85, 91	sylbid 230	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = ({𝑘} × (𝐾‘𝑘)) → 𝑛 = 𝑘))
93	83, 92	sylbid 230	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = (𝐴‘𝑘) → 𝑛 = 𝑘))
94	93	rgen2a 2977	. . . 4 ⊢ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴‘𝑛) = (𝐴‘𝑘) → 𝑛 = 𝑘)
95		dff13 6512	. . . 4 ⊢ (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴‘𝑛) = (𝐴‘𝑘) → 𝑛 = 𝑘)))
96	77, 94, 95	mpbir2an 955	. . 3 ⊢ 𝐴:ω–1-1→V
97		f1f1orn 6148	. . . 4 ⊢ (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
98	64	f1oen 7976	. . . 4 ⊢ (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
99		ensym 8005	. . . 4 ⊢ (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
100	97, 98, 99	3syl 18	. . 3 ⊢ (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
101	7	rneqi 5352	. . . . 5 ⊢ ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
102		dmmptg 5632	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ({𝑛} × (𝐾‘𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) = ω)
103	6	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ω → ({𝑛} × (𝐾‘𝑛)) ∈ V)
104	102, 103	mprg 2926	. . . . . . 7 ⊢ dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) = ω
105	104, 64	eqeltri 2697	. . . . . 6 ⊢ dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V
106		funmpt 5926	. . . . . 6 ⊢ Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
107		funrnex 7133	. . . . . 6 ⊢ (dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V → (Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) → ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V))
108	105, 106, 107	mp2 9	. . . . 5 ⊢ ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V
109	101, 108	eqeltri 2697	. . . 4 ⊢ ran 𝐴 ∈ V
110		breq1 4656	. . . . 5 ⊢ (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
111		raleq 3138	. . . . . 6 ⊢ (𝑎 = ran 𝐴 → (∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
112	111	exbidv 1850	. . . . 5 ⊢ (𝑎 = ran 𝐴 → (∃𝑓∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
113	110, 112	imbi12d 334	. . . 4 ⊢ (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))))
114		ax-cc 9257	. . . 4 ⊢ (𝑎 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
115	109, 113, 114	vtocl 3259	. . 3 ⊢ (ran 𝐴 ≈ ω → ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
116	96, 100, 115	mp2b 10	. 2 ⊢ ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
117	74, 116	exlimiiv 1859	1 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∅c0 3915 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ran crn 5115 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 ωcom 7065 2^nd c2nd 7167 ≈ cen 7952

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-reu 2919 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-2nd 7169 df-er 7742 df-en 7956

This theorem is referenced by: axcc2 9259

W3C validator