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Theorem axcc2lem 9114
Description: Lemma for axcc2 9115. (Contributed by Mario Carneiro, 8-Feb-2013.)
Hypotheses
Ref Expression
axcc2lem.1 𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
axcc2lem.2 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
axcc2lem.3 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))
Assertion
Ref Expression
axcc2lem 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
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.
StepHypRef Expression
1 fvex 6094 . . . 4 (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V
2 axcc2lem.3 . . . 4 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))
31, 2fnmpti 5917 . . 3 𝐺 Fn ω
4 snex 4826 . . . . . . . . . . . . . . 15 {𝑛} ∈ V
5 fvex 6094 . . . . . . . . . . . . . . 15 (𝐾𝑛) ∈ V
64, 5xpex 6833 . . . . . . . . . . . . . 14 ({𝑛} × (𝐾𝑛)) ∈ V
7 axcc2lem.2 . . . . . . . . . . . . . . 15 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
87fvmpt2 6181 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾𝑛)) ∈ V) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
96, 8mpan2 702 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
10 vex 3171 . . . . . . . . . . . . . . 15 𝑛 ∈ V
1110snnz 4247 . . . . . . . . . . . . . 14 {𝑛} ≠ ∅
12 0ex 4709 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
1312snnz 4247 . . . . . . . . . . . . . . . . 17 {∅} ≠ ∅
14 iftrue 4037 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = {∅})
1514neeq1d 2836 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) = ∅ → (if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
1613, 15mpbiri 246 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
17 iffalse 4040 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
18 df-ne 2777 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) ≠ ∅ ↔ ¬ (𝐹𝑛) = ∅)
1918biimpri 216 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → (𝐹𝑛) ≠ ∅)
2017, 19eqnetrd 2844 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
2116, 20pm2.61i 174 . . . . . . . . . . . . . . 15 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅
22 p0ex 4770 . . . . . . . . . . . . . . . . . 18 {∅} ∈ V
23 fvex 6094 . . . . . . . . . . . . . . . . . 18 (𝐹𝑛) ∈ V
2422, 23ifex 4101 . . . . . . . . . . . . . . . . 17 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V
25 axcc2lem.1 . . . . . . . . . . . . . . . . . 18 𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2625fvmpt2 6181 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2724, 26mpan2 702 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ω → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2827neeq1d 2836 . . . . . . . . . . . . . . 15 (𝑛 ∈ ω → ((𝐾𝑛) ≠ ∅ ↔ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅))
2921, 28mpbiri 246 . . . . . . . . . . . . . 14 (𝑛 ∈ ω → (𝐾𝑛) ≠ ∅)
30 xpnz 5454 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾𝑛)) ≠ ∅)
3130biimpi 204 . . . . . . . . . . . . . 14 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → ({𝑛} × (𝐾𝑛)) ≠ ∅)
3211, 29, 31sylancr 693 . . . . . . . . . . . . 13 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ≠ ∅)
339, 32eqnetrd 2844 . . . . . . . . . . . 12 (𝑛 ∈ ω → (𝐴𝑛) ≠ ∅)
346, 7fnmpti 5917 . . . . . . . . . . . . . 14 𝐴 Fn ω
35 fnfvelrn 6245 . . . . . . . . . . . . . 14 ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴𝑛) ∈ ran 𝐴)
3634, 35mpan 701 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) ∈ ran 𝐴)
37 neeq1 2839 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → (𝑧 ≠ ∅ ↔ (𝐴𝑛) ≠ ∅))
38 fveq2 6084 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → (𝑓𝑧) = (𝑓‘(𝐴𝑛)))
39 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → 𝑧 = (𝐴𝑛))
4038, 39eleq12d 2677 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4137, 40imbi12d 332 . . . . . . . . . . . . . 14 (𝑧 = (𝐴𝑛) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4241rspccv 3274 . . . . . . . . . . . . 13 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝐴𝑛) ∈ ran 𝐴 → ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4336, 42syl5 33 . . . . . . . . . . . 12 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4433, 43mpdi 43 . . . . . . . . . . 11 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4544impcom 444 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))
469eleq2d 2668 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4746adantr 479 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4845, 47mpbid 220 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛)))
49 xp2nd 7063 . . . . . . . . 9 ((𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛)) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
5048, 49syl 17 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
51503adant3 1073 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
522fvmpt2 6181 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
531, 52mpan2 702 . . . . . . . . 9 (𝑛 ∈ ω → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
54533ad2ant1 1074 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
5554eqcomd 2611 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) = (𝐺𝑛))
56273ad2ant1 1074 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
57 ifnefalse 4043 . . . . . . . . 9 ((𝐹𝑛) ≠ ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
58573ad2ant3 1076 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
5956, 58eqtrd 2639 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = (𝐹𝑛))
6051, 55, 593eltr3d 2697 . . . . . 6 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) ∈ (𝐹𝑛))
61603expia 1258 . . . . 5 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
6261expcom 449 . . . 4 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
6362ralrimiv 2943 . . 3 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
64 omex 8396 . . . . 5 ω ∈ V
65 fnex 6360 . . . . 5 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
663, 64, 65mp2an 703 . . . 4 𝐺 ∈ V
67 fneq1 5875 . . . . 5 (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
68 fveq1 6083 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
6968eleq1d 2667 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝐺𝑛) ∈ (𝐹𝑛)))
7069imbi2d 328 . . . . . 6 (𝑔 = 𝐺 → (((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7170ralbidv 2964 . . . . 5 (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7267, 71anbi12d 742 . . . 4 (𝑔 = 𝐺 → ((𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))) ↔ (𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))))
7366, 72spcev 3268 . . 3 ((𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))))
743, 63, 73sylancr 693 . 2 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))))
756a1i 11 . . . . . 6 ((ω ∈ V ∧ 𝑛 ∈ ω) → ({𝑛} × (𝐾𝑛)) ∈ V)
7675, 7fmptd 6273 . . . . 5 (ω ∈ V → 𝐴:ω⟶V)
7764, 76ax-mp 5 . . . 4 𝐴:ω⟶V
78 sneq 4130 . . . . . . . . . 10 (𝑛 = 𝑘 → {𝑛} = {𝑘})
79 fveq2 6084 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐾𝑛) = (𝐾𝑘))
8078, 79xpeq12d 5050 . . . . . . . . 9 (𝑛 = 𝑘 → ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)))
8180, 7, 6fvmpt3i 6177 . . . . . . . 8 (𝑘 ∈ ω → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8281adantl 480 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8382eqeq2d 2615 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) ↔ (𝐴𝑛) = ({𝑘} × (𝐾𝑘))))
849adantr 479 . . . . . . . 8 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
8584eqeq1d 2607 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘))))
86 xp11 5470 . . . . . . . . . 10 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8711, 29, 86sylancr 693 . . . . . . . . 9 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8810sneqr 4302 . . . . . . . . . 10 ({𝑛} = {𝑘} → 𝑛 = 𝑘)
8988adantr 479 . . . . . . . . 9 (({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘)) → 𝑛 = 𝑘)
9087, 89syl6bi 241 . . . . . . . 8 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9190adantr 479 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9285, 91sylbid 228 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9383, 92sylbid 228 . . . . 5 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘))
9493rgen2a 2955 . . . 4 𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)
95 dff13 6390 . . . 4 (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)))
9677, 94, 95mpbir2an 956 . . 3 𝐴:ω–1-1→V
97 f1f1orn 6042 . . . 4 (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
9864f1oen 7835 . . . 4 (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
99 ensym 7864 . . . 4 (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
10097, 98, 993syl 18 . . 3 (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
1017rneqi 5256 . . . . 5 ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
102 dmmptg 5531 . . . . . . . 8 (∀𝑛 ∈ ω ({𝑛} × (𝐾𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω)
1036a1i 11 . . . . . . . 8 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ∈ V)
104102, 103mprg 2905 . . . . . . 7 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω
105104, 64eqeltri 2679 . . . . . 6 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
106 funmpt 5822 . . . . . 6 Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
107 funrnex 6999 . . . . . 6 (dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V → (Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) → ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V))
108105, 106, 107mp2 9 . . . . 5 ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
109101, 108eqeltri 2679 . . . 4 ran 𝐴 ∈ V
110 breq1 4576 . . . . 5 (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
111 raleq 3110 . . . . . 6 (𝑎 = ran 𝐴 → (∀𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
112111exbidv 1835 . . . . 5 (𝑎 = ran 𝐴 → (∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
113110, 112imbi12d 332 . . . 4 (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
114 ax-cc 9113 . . . 4 (𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
115109, 113, 114vtocl 3227 . . 3 (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
11696, 100, 115mp2b 10 . 2 𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
11774, 116exlimiiv 1844 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wne 2775  wral 2891  Vcvv 3168  c0 3869  ifcif 4031  {csn 4120   class class class wbr 4573  cmpt 4633   × cxp 5022  dom cdm 5024  ran crn 5025  Fun wfun 5780   Fn wfn 5781  wf 5782  1-1wf1 5783  1-1-ontowf1o 5785  cfv 5786  ωcom 6930  2nd c2nd 7031  cen 7811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cc 9113
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-2nd 7033  df-er 7602  df-en 7815
This theorem is referenced by:  axcc2  9115
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