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Theorem axcc2lem 9218
 Description: Lemma for axcc2 9219. (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 6168 . . . 4 (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V
2 axcc2lem.3 . . . 4 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))
31, 2fnmpti 5989 . . 3 𝐺 Fn ω
4 snex 4879 . . . . . . . . . . . . . . 15 {𝑛} ∈ V
5 fvex 6168 . . . . . . . . . . . . . . 15 (𝐾𝑛) ∈ V
64, 5xpex 6927 . . . . . . . . . . . . . 14 ({𝑛} × (𝐾𝑛)) ∈ V
7 axcc2lem.2 . . . . . . . . . . . . . . 15 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
87fvmpt2 6258 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾𝑛)) ∈ V) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
96, 8mpan2 706 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
10 vex 3193 . . . . . . . . . . . . . . 15 𝑛 ∈ V
1110snnz 4286 . . . . . . . . . . . . . 14 {𝑛} ≠ ∅
12 0ex 4760 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
1312snnz 4286 . . . . . . . . . . . . . . . . 17 {∅} ≠ ∅
14 iftrue 4070 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = {∅})
1514neeq1d 2849 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) = ∅ → (if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
1613, 15mpbiri 248 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
17 iffalse 4073 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
18 df-ne 2791 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) ≠ ∅ ↔ ¬ (𝐹𝑛) = ∅)
1918biimpri 218 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → (𝐹𝑛) ≠ ∅)
2017, 19eqnetrd 2857 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
2116, 20pm2.61i 176 . . . . . . . . . . . . . . 15 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅
22 p0ex 4823 . . . . . . . . . . . . . . . . . 18 {∅} ∈ V
23 fvex 6168 . . . . . . . . . . . . . . . . . 18 (𝐹𝑛) ∈ V
2422, 23ifex 4134 . . . . . . . . . . . . . . . . 17 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V
25 axcc2lem.1 . . . . . . . . . . . . . . . . . 18 𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2625fvmpt2 6258 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2724, 26mpan2 706 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ω → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2827neeq1d 2849 . . . . . . . . . . . . . . 15 (𝑛 ∈ ω → ((𝐾𝑛) ≠ ∅ ↔ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅))
2921, 28mpbiri 248 . . . . . . . . . . . . . 14 (𝑛 ∈ ω → (𝐾𝑛) ≠ ∅)
30 xpnz 5522 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾𝑛)) ≠ ∅)
3130biimpi 206 . . . . . . . . . . . . . 14 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → ({𝑛} × (𝐾𝑛)) ≠ ∅)
3211, 29, 31sylancr 694 . . . . . . . . . . . . 13 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ≠ ∅)
339, 32eqnetrd 2857 . . . . . . . . . . . 12 (𝑛 ∈ ω → (𝐴𝑛) ≠ ∅)
346, 7fnmpti 5989 . . . . . . . . . . . . . 14 𝐴 Fn ω
35 fnfvelrn 6322 . . . . . . . . . . . . . 14 ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴𝑛) ∈ ran 𝐴)
3634, 35mpan 705 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) ∈ ran 𝐴)
37 neeq1 2852 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → (𝑧 ≠ ∅ ↔ (𝐴𝑛) ≠ ∅))
38 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → (𝑓𝑧) = (𝑓‘(𝐴𝑛)))
39 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → 𝑧 = (𝐴𝑛))
4038, 39eleq12d 2692 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4137, 40imbi12d 334 . . . . . . . . . . . . . 14 (𝑧 = (𝐴𝑛) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4241rspccv 3296 . . . . . . . . . . . . 13 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝐴𝑛) ∈ ran 𝐴 → ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4336, 42syl5 34 . . . . . . . . . . . 12 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4433, 43mpdi 45 . . . . . . . . . . 11 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4544impcom 446 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))
469eleq2d 2684 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4746adantr 481 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4845, 47mpbid 222 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛)))
49 xp2nd 7159 . . . . . . . . 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 1079 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
522fvmpt2 6258 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
531, 52mpan2 706 . . . . . . . . 9 (𝑛 ∈ ω → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
54533ad2ant1 1080 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
5554eqcomd 2627 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) = (𝐺𝑛))
56273ad2ant1 1080 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
57 ifnefalse 4076 . . . . . . . . 9 ((𝐹𝑛) ≠ ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
58573ad2ant3 1082 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
5956, 58eqtrd 2655 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = (𝐹𝑛))
6051, 55, 593eltr3d 2712 . . . . . 6 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) ∈ (𝐹𝑛))
61603expia 1264 . . . . 5 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
6261expcom 451 . . . 4 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
6362ralrimiv 2961 . . 3 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
64 omex 8500 . . . . 5 ω ∈ V
65 fnex 6446 . . . . 5 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
663, 64, 65mp2an 707 . . . 4 𝐺 ∈ V
67 fneq1 5947 . . . . 5 (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
68 fveq1 6157 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
6968eleq1d 2683 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝐺𝑛) ∈ (𝐹𝑛)))
7069imbi2d 330 . . . . . 6 (𝑔 = 𝐺 → (((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7170ralbidv 2982 . . . . 5 (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7267, 71anbi12d 746 . . . 4 (𝑔 = 𝐺 → ((𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))) ↔ (𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))))
7366, 72spcev 3290 . . 3 ((𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))))
743, 63, 73sylancr 694 . 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 6351 . . . . 5 (ω ∈ V → 𝐴:ω⟶V)
7764, 76ax-mp 5 . . . 4 𝐴:ω⟶V
78 sneq 4165 . . . . . . . . . 10 (𝑛 = 𝑘 → {𝑛} = {𝑘})
79 fveq2 6158 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐾𝑛) = (𝐾𝑘))
8078, 79xpeq12d 5110 . . . . . . . . 9 (𝑛 = 𝑘 → ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)))
8180, 7, 6fvmpt3i 6254 . . . . . . . 8 (𝑘 ∈ ω → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8281adantl 482 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8382eqeq2d 2631 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) ↔ (𝐴𝑛) = ({𝑘} × (𝐾𝑘))))
849adantr 481 . . . . . . . 8 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
8584eqeq1d 2623 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘))))
86 xp11 5538 . . . . . . . . . 10 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8711, 29, 86sylancr 694 . . . . . . . . 9 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8810sneqr 4346 . . . . . . . . . 10 ({𝑛} = {𝑘} → 𝑛 = 𝑘)
8988adantr 481 . . . . . . . . 9 (({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘)) → 𝑛 = 𝑘)
9087, 89syl6bi 243 . . . . . . . 8 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9190adantr 481 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9285, 91sylbid 230 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9383, 92sylbid 230 . . . . 5 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘))
9493rgen2a 2973 . . . 4 𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)
95 dff13 6477 . . . 4 (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)))
9677, 94, 95mpbir2an 954 . . 3 𝐴:ω–1-1→V
97 f1f1orn 6115 . . . 4 (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
9864f1oen 7936 . . . 4 (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
99 ensym 7965 . . . 4 (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
10097, 98, 993syl 18 . . 3 (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
1017rneqi 5322 . . . . 5 ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
102 dmmptg 5601 . . . . . . . 8 (∀𝑛 ∈ ω ({𝑛} × (𝐾𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω)
1036a1i 11 . . . . . . . 8 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ∈ V)
104102, 103mprg 2922 . . . . . . 7 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω
105104, 64eqeltri 2694 . . . . . 6 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
106 funmpt 5894 . . . . . 6 Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
107 funrnex 7095 . . . . . 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 2694 . . . 4 ran 𝐴 ∈ V
110 breq1 4626 . . . . 5 (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
111 raleq 3131 . . . . . 6 (𝑎 = ran 𝐴 → (∀𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
112111exbidv 1847 . . . . 5 (𝑎 = ran 𝐴 → (∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
113110, 112imbi12d 334 . . . 4 (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
114 ax-cc 9217 . . . 4 (𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
115109, 113, 114vtocl 3249 . . 3 (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
11696, 100, 115mp2b 10 . 2 𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
11774, 116exlimiiv 1856 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2908  Vcvv 3190  ∅c0 3897  ifcif 4064  {csn 4155   class class class wbr 4623   ↦ cmpt 4683   × cxp 5082  dom cdm 5084  ran crn 5085  Fun wfun 5851   Fn wfn 5852  ⟶wf 5853  –1-1→wf1 5854  –1-1-onto→wf1o 5856  ‘cfv 5857  ωcom 7027  2nd c2nd 7127   ≈ cen 7912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cc 9217 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-2nd 7129  df-er 7702  df-en 7916 This theorem is referenced by:  axcc2  9219
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