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Theorem axcc2lem 9296
Description: Lemma for axcc2 9297. (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 6239 . . . 4 (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V
2 axcc2lem.3 . . . 4 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))
31, 2fnmpti 6060 . . 3 𝐺 Fn ω
4 snex 4938 . . . . . . . . . . . . . . 15 {𝑛} ∈ V
5 fvex 6239 . . . . . . . . . . . . . . 15 (𝐾𝑛) ∈ V
64, 5xpex 7004 . . . . . . . . . . . . . 14 ({𝑛} × (𝐾𝑛)) ∈ V
7 axcc2lem.2 . . . . . . . . . . . . . . 15 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
87fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾𝑛)) ∈ V) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
96, 8mpan2 707 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
10 vex 3234 . . . . . . . . . . . . . . 15 𝑛 ∈ V
1110snnz 4340 . . . . . . . . . . . . . 14 {𝑛} ≠ ∅
12 0ex 4823 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
1312snnz 4340 . . . . . . . . . . . . . . . . 17 {∅} ≠ ∅
14 iftrue 4125 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = {∅})
1514neeq1d 2882 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) = ∅ → (if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
1613, 15mpbiri 248 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
17 iffalse 4128 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
18 df-ne 2824 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) ≠ ∅ ↔ ¬ (𝐹𝑛) = ∅)
1918biimpri 218 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → (𝐹𝑛) ≠ ∅)
2017, 19eqnetrd 2890 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
2116, 20pm2.61i 176 . . . . . . . . . . . . . . 15 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅
22 p0ex 4883 . . . . . . . . . . . . . . . . . 18 {∅} ∈ V
23 fvex 6239 . . . . . . . . . . . . . . . . . 18 (𝐹𝑛) ∈ V
2422, 23ifex 4189 . . . . . . . . . . . . . . . . 17 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V
25 axcc2lem.1 . . . . . . . . . . . . . . . . . 18 𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2625fvmpt2 6330 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2724, 26mpan2 707 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ω → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2827neeq1d 2882 . . . . . . . . . . . . . . 15 (𝑛 ∈ ω → ((𝐾𝑛) ≠ ∅ ↔ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅))
2921, 28mpbiri 248 . . . . . . . . . . . . . 14 (𝑛 ∈ ω → (𝐾𝑛) ≠ ∅)
30 xpnz 5588 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾𝑛)) ≠ ∅)
3130biimpi 206 . . . . . . . . . . . . . 14 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → ({𝑛} × (𝐾𝑛)) ≠ ∅)
3211, 29, 31sylancr 696 . . . . . . . . . . . . 13 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ≠ ∅)
339, 32eqnetrd 2890 . . . . . . . . . . . 12 (𝑛 ∈ ω → (𝐴𝑛) ≠ ∅)
346, 7fnmpti 6060 . . . . . . . . . . . . . 14 𝐴 Fn ω
35 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴𝑛) ∈ ran 𝐴)
3634, 35mpan 706 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) ∈ ran 𝐴)
37 neeq1 2885 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → (𝑧 ≠ ∅ ↔ (𝐴𝑛) ≠ ∅))
38 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → (𝑓𝑧) = (𝑓‘(𝐴𝑛)))
39 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → 𝑧 = (𝐴𝑛))
4038, 39eleq12d 2724 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4137, 40imbi12d 333 . . . . . . . . . . . . . 14 (𝑧 = (𝐴𝑛) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4241rspccv 3337 . . . . . . . . . . . . 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 445 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))
469eleq2d 2716 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4746adantr 480 . . . . . . . . . 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 7243 . . . . . . . . 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 1101 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
522fvmpt2 6330 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
531, 52mpan2 707 . . . . . . . . 9 (𝑛 ∈ ω → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
54533ad2ant1 1102 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
5554eqcomd 2657 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) = (𝐺𝑛))
56273ad2ant1 1102 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
57 ifnefalse 4131 . . . . . . . . 9 ((𝐹𝑛) ≠ ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
58573ad2ant3 1104 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
5956, 58eqtrd 2685 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = (𝐹𝑛))
6051, 55, 593eltr3d 2744 . . . . . 6 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) ∈ (𝐹𝑛))
61603expia 1286 . . . . 5 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
6261expcom 450 . . . 4 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
6362ralrimiv 2994 . . 3 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
64 omex 8578 . . . . 5 ω ∈ V
65 fnex 6522 . . . . 5 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
663, 64, 65mp2an 708 . . . 4 𝐺 ∈ V
67 fneq1 6017 . . . . 5 (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
68 fveq1 6228 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
6968eleq1d 2715 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝐺𝑛) ∈ (𝐹𝑛)))
7069imbi2d 329 . . . . . 6 (𝑔 = 𝐺 → (((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7170ralbidv 3015 . . . . 5 (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7267, 71anbi12d 747 . . . 4 (𝑔 = 𝐺 → ((𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))) ↔ (𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))))
7366, 72spcev 3331 . . 3 ((𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))))
743, 63, 73sylancr 696 . 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 6425 . . . . 5 (ω ∈ V → 𝐴:ω⟶V)
7764, 76ax-mp 5 . . . 4 𝐴:ω⟶V
78 sneq 4220 . . . . . . . . . 10 (𝑛 = 𝑘 → {𝑛} = {𝑘})
79 fveq2 6229 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐾𝑛) = (𝐾𝑘))
8078, 79xpeq12d 5174 . . . . . . . . 9 (𝑛 = 𝑘 → ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)))
8180, 7, 6fvmpt3i 6326 . . . . . . . 8 (𝑘 ∈ ω → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8281adantl 481 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8382eqeq2d 2661 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) ↔ (𝐴𝑛) = ({𝑘} × (𝐾𝑘))))
849adantr 480 . . . . . . . 8 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
8584eqeq1d 2653 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘))))
86 xp11 5604 . . . . . . . . . 10 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8711, 29, 86sylancr 696 . . . . . . . . 9 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8810sneqr 4403 . . . . . . . . . 10 ({𝑛} = {𝑘} → 𝑛 = 𝑘)
8988adantr 480 . . . . . . . . 9 (({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘)) → 𝑛 = 𝑘)
9087, 89syl6bi 243 . . . . . . . 8 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9190adantr 480 . . . . . . 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 3006 . . . 4 𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)
95 dff13 6552 . . . 4 (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)))
9677, 94, 95mpbir2an 975 . . 3 𝐴:ω–1-1→V
97 f1f1orn 6186 . . . 4 (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
9864f1oen 8018 . . . 4 (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
99 ensym 8046 . . . 4 (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
10097, 98, 993syl 18 . . 3 (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
1017rneqi 5384 . . . . 5 ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
102 dmmptg 5670 . . . . . . . 8 (∀𝑛 ∈ ω ({𝑛} × (𝐾𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω)
1036a1i 11 . . . . . . . 8 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ∈ V)
104102, 103mprg 2955 . . . . . . 7 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω
105104, 64eqeltri 2726 . . . . . 6 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
106 funmpt 5964 . . . . . 6 Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
107 funrnex 7175 . . . . . 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 2726 . . . 4 ran 𝐴 ∈ V
110 breq1 4688 . . . . 5 (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
111 raleq 3168 . . . . . 6 (𝑎 = ran 𝐴 → (∀𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
112111exbidv 1890 . . . . 5 (𝑎 = ran 𝐴 → (∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
113110, 112imbi12d 333 . . . 4 (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
114 ax-cc 9295 . . . 4 (𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
115109, 113, 114vtocl 3290 . . 3 (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
11696, 100, 115mp2b 10 . 2 𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
11774, 116exlimiiv 1899 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  Vcvv 3231  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  ωcom 7107  2nd c2nd 7209  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-2nd 7211  df-er 7787  df-en 7998
This theorem is referenced by:  axcc2  9297
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