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Theorem axcc2lem 10337

Description: Lemma for axcc2 10338. (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 6844	. . . 4 ⊢ (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ V
2		axcc2lem.3	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ω ↦ (2^nd ‘(𝑓‘(𝐴‘𝑛))))
3	1, 2	fnmpti 6632	. . 3 ⊢ 𝐺 Fn ω
4		vsnex 5376	. . . . . . . . . . . . . . 15 ⊢ {𝑛} ∈ V
5		fvex 6844	. . . . . . . . . . . . . . 15 ⊢ (𝐾‘𝑛) ∈ V
6	4, 5	xpex 7695	. . . . . . . . . . . . . 14 ⊢ ({𝑛} × (𝐾‘𝑛)) ∈ V
7		axcc2lem.2	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
8	7	fvmpt2 6949	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾‘𝑛)) ∈ V) → (𝐴‘𝑛) = ({𝑛} × (𝐾‘𝑛)))
9	6, 8	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → (𝐴‘𝑛) = ({𝑛} × (𝐾‘𝑛)))
10		vex 3442	. . . . . . . . . . . . . . 15 ⊢ 𝑛 ∈ V
11	10	snnz 4730	. . . . . . . . . . . . . 14 ⊢ {𝑛} ≠ ∅
12		0ex 5249	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
13	12	snnz 4730	. . . . . . . . . . . . . . . . 17 ⊢ {∅} ≠ ∅
14		iftrue 4482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = {∅})
15	14	neeq1d 2989	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑛) = ∅ → (if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
16	13, 15	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅)
17		iffalse 4485	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = (𝐹‘𝑛))
18		neqne 2938	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝐹‘𝑛) = ∅ → (𝐹‘𝑛) ≠ ∅)
19	17, 18	eqnetrd 2997	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐹‘𝑛) = ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅)
20	16, 19	pm2.61i 182	. . . . . . . . . . . . . . 15 ⊢ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅
21		p0ex 5326	. . . . . . . . . . . . . . . . . 18 ⊢ {∅} ∈ V
22		fvex 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑛) ∈ V
23	21, 22	ifex 4527	. . . . . . . . . . . . . . . . 17 ⊢ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ∈ V
24		axcc2lem.1	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐾 = (𝑛 ∈ ω ↦ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
25	24	fvmpt2 6949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ∈ V) → (𝐾‘𝑛) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
26	23, 25	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → (𝐾‘𝑛) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
27	26	neeq1d 2989	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → ((𝐾‘𝑛) ≠ ∅ ↔ if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) ≠ ∅))
28	20, 27	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → (𝐾‘𝑛) ≠ ∅)
29		xpnz 6114	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ≠ ∅ ∧ (𝐾‘𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾‘𝑛)) ≠ ∅)
30	29	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (({𝑛} ≠ ∅ ∧ (𝐾‘𝑛) ≠ ∅) → ({𝑛} × (𝐾‘𝑛)) ≠ ∅)
31	11, 28, 30	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → ({𝑛} × (𝐾‘𝑛)) ≠ ∅)
32	9, 31	eqnetrd 2997	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → (𝐴‘𝑛) ≠ ∅)
33	6, 7	fnmpti 6632	. . . . . . . . . . . . . 14 ⊢ 𝐴 Fn ω
34		fnfvelrn 7022	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴‘𝑛) ∈ ran 𝐴)
35	33, 34	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → (𝐴‘𝑛) ∈ ran 𝐴)
36		neeq1 2992	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑛) → (𝑧 ≠ ∅ ↔ (𝐴‘𝑛) ≠ ∅))
37		fveq2 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑛) → (𝑓‘𝑧) = (𝑓‘(𝐴‘𝑛)))
38		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝐴‘𝑛) → 𝑧 = (𝐴‘𝑛))
39	37, 38	eleq12d 2827	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐴‘𝑛) → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛)))
40	36, 39	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝐴‘𝑛) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ((𝐴‘𝑛) ≠ ∅ → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))))
41	40	rspccv 3571	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ((𝐴‘𝑛) ∈ ran 𝐴 → ((𝐴‘𝑛) ≠ ∅ → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))))
42	35, 41	syl5 34	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐴‘𝑛) ≠ ∅ → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))))
43	32, 42	mpdi 45	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑛 ∈ ω → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛)))
44	43	impcom 407	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛))
45	9	eleq2d 2819	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛) ↔ (𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛))))
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝑓‘(𝐴‘𝑛)) ∈ (𝐴‘𝑛) ↔ (𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛))))
47	44, 46	mpbid 232	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛)))
48		xp2nd 7963	. . . . . . . . 9 ⊢ ((𝑓‘(𝐴‘𝑛)) ∈ ({𝑛} × (𝐾‘𝑛)) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ (𝐾‘𝑛))
49	47, 48	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ (𝐾‘𝑛))
50	49	3adant3 1132	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ (𝐾‘𝑛))
51	2	fvmpt2 6949	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ (2^nd ‘(𝑓‘(𝐴‘𝑛))) ∈ V) → (𝐺‘𝑛) = (2^nd ‘(𝑓‘(𝐴‘𝑛))))
52	1, 51	mpan2 691	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (𝐺‘𝑛) = (2^nd ‘(𝑓‘(𝐴‘𝑛))))
53	52	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐺‘𝑛) = (2^nd ‘(𝑓‘(𝐴‘𝑛))))
54	53	eqcomd 2739	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (2^nd ‘(𝑓‘(𝐴‘𝑛))) = (𝐺‘𝑛))
55	26	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐾‘𝑛) = if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)))
56		ifnefalse 4488	. . . . . . . . 9 ⊢ ((𝐹‘𝑛) ≠ ∅ → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = (𝐹‘𝑛))
57	56	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → if((𝐹‘𝑛) = ∅, {∅}, (𝐹‘𝑛)) = (𝐹‘𝑛))
58	55, 57	eqtrd 2768	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐾‘𝑛) = (𝐹‘𝑛))
59	50, 54, 58	3eltr3d 2847	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ∧ (𝐹‘𝑛) ≠ ∅) → (𝐺‘𝑛) ∈ (𝐹‘𝑛))
60	59	3expia 1121	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛)))
61	60	expcom 413	. . . 4 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))))
62	61	ralrimiv 3125	. . 3 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛)))
63		omex 9543	. . . . 5 ⊢ ω ∈ V
64		fnex 7160	. . . . 5 ⊢ ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
65	3, 63, 64	mp2an 692	. . . 4 ⊢ 𝐺 ∈ V
66		fneq1 6580	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
67		fveq1 6830	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑛) = (𝐺‘𝑛))
68	67	eleq1d 2818	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑛) ∈ (𝐹‘𝑛) ↔ (𝐺‘𝑛) ∈ (𝐹‘𝑛)))
69	68	imbi2d 340	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ↔ ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))))
70	69	ralbidv 3157	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))))
71	66, 70	anbi12d 632	. . . 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)))))
72	65, 71	spcev 3558	. . 3 ⊢ ((𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝐺‘𝑛) ∈ (𝐹‘𝑛))) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛))))
73	3, 62, 72	sylancr 587	. 2 ⊢ (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹‘𝑛) ≠ ∅ → (𝑔‘𝑛) ∈ (𝐹‘𝑛))))
74	6	a1i 11	. . . . . 6 ⊢ ((ω ∈ V ∧ 𝑛 ∈ ω) → ({𝑛} × (𝐾‘𝑛)) ∈ V)
75	74, 7	fmptd 7056	. . . . 5 ⊢ (ω ∈ V → 𝐴:ω⟶V)
76	63, 75	ax-mp 5	. . . 4 ⊢ 𝐴:ω⟶V
77		sneq 4587	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘})
78		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝐾‘𝑛) = (𝐾‘𝑘))
79	77, 78	xpeq12d 5652	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)))
80	79, 7, 6	fvmpt3i 6943	. . . . . . . 8 ⊢ (𝑘 ∈ ω → (𝐴‘𝑘) = ({𝑘} × (𝐾‘𝑘)))
81	80	adantl 481	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴‘𝑘) = ({𝑘} × (𝐾‘𝑘)))
82	81	eqeq2d 2744	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = (𝐴‘𝑘) ↔ (𝐴‘𝑛) = ({𝑘} × (𝐾‘𝑘))))
83	9	adantr 480	. . . . . . . 8 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴‘𝑛) = ({𝑛} × (𝐾‘𝑛)))
84	83	eqeq1d 2735	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = ({𝑘} × (𝐾‘𝑘)) ↔ ({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘))))
85		xp11 6130	. . . . . . . . . 10 ⊢ (({𝑛} ≠ ∅ ∧ (𝐾‘𝑛) ≠ ∅) → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾‘𝑛) = (𝐾‘𝑘))))
86	11, 28, 85	sylancr 587	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾‘𝑛) = (𝐾‘𝑘))))
87	10	sneqr 4793	. . . . . . . . . 10 ⊢ ({𝑛} = {𝑘} → 𝑛 = 𝑘)
88	87	adantr 480	. . . . . . . . 9 ⊢ (({𝑛} = {𝑘} ∧ (𝐾‘𝑛) = (𝐾‘𝑘)) → 𝑛 = 𝑘)
89	86, 88	biimtrdi 253	. . . . . . . 8 ⊢ (𝑛 ∈ ω → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) → 𝑛 = 𝑘))
90	89	adantr 480	. . . . . . 7 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (({𝑛} × (𝐾‘𝑛)) = ({𝑘} × (𝐾‘𝑘)) → 𝑛 = 𝑘))
91	84, 90	sylbid 240	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = ({𝑘} × (𝐾‘𝑘)) → 𝑛 = 𝑘))
92	82, 91	sylbid 240	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴‘𝑛) = (𝐴‘𝑘) → 𝑛 = 𝑘))
93	92	rgen2 3174	. . . 4 ⊢ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴‘𝑛) = (𝐴‘𝑘) → 𝑛 = 𝑘)
94		dff13 7197	. . . 4 ⊢ (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴‘𝑛) = (𝐴‘𝑘) → 𝑛 = 𝑘)))
95	76, 93, 94	mpbir2an 711	. . 3 ⊢ 𝐴:ω–1-1→V
96		f1f1orn 6782	. . . 4 ⊢ (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
97	63	f1oen 8904	. . . 4 ⊢ (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
98		ensym 8935	. . . 4 ⊢ (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
99	96, 97, 98	3syl 18	. . 3 ⊢ (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
100	7	rneqi 5884	. . . . 5 ⊢ ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
101		dmmptg 6197	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ({𝑛} × (𝐾‘𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) = ω)
102	6	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ω → ({𝑛} × (𝐾‘𝑛)) ∈ V)
103	101, 102	mprg 3055	. . . . . . 7 ⊢ dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) = ω
104	103, 63	eqeltri 2829	. . . . . 6 ⊢ dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V
105		funmpt 6527	. . . . . 6 ⊢ Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛)))
106		funrnex 7895	. . . . . 6 ⊢ (dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V → (Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) → ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V))
107	104, 105, 106	mp2 9	. . . . 5 ⊢ ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾‘𝑛))) ∈ V
108	100, 107	eqeltri 2829	. . . 4 ⊢ ran 𝐴 ∈ V
109		breq1 5098	. . . . 5 ⊢ (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
110		raleq 3291	. . . . . 6 ⊢ (𝑎 = ran 𝐴 → (∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
111	110	exbidv 1922	. . . . 5 ⊢ (𝑎 = ran 𝐴 → (∃𝑓∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
112	109, 111	imbi12d 344	. . . 4 ⊢ (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))))
113		ax-cc 10336	. . . 4 ⊢ (𝑎 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑎 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
114	108, 112, 113	vtocl 3513	. . 3 ⊢ (ran 𝐴 ≈ ω → ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
115	95, 99, 114	mp2b 10	. 2 ⊢ ∃𝑓∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)
116	73, 115	exlimiiv 1932	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 Vcvv 3438 ∅c0 4284 ifcif 4476 {csn 4577 class class class wbr 5095 ↦ cmpt 5176 × cxp 5619 dom cdm 5621 ran crn 5622 Fun wfun 6483 Fn wfn 6484 ⟶wf 6485 –1-1→wf1 6486 –1-1-onto→wf1o 6488 ‘cfv 6489 ωcom 7805 2^nd c2nd 7929 ≈ cen 8875

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 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9541 ax-cc 10336

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 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 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 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-om 7806 df-2nd 7931 df-er 8631 df-en 8879

This theorem is referenced by: axcc2 10338

W3C validator