MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axcc2lem Structured version   Visualization version   GIF version

Theorem axcc2lem 9847
Description: Lemma for axcc2 9848. (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 6680 . . . 4 (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V
2 axcc2lem.3 . . . 4 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))
31, 2fnmpti 6488 . . 3 𝐺 Fn ω
4 snex 5328 . . . . . . . . . . . . . . 15 {𝑛} ∈ V
5 fvex 6680 . . . . . . . . . . . . . . 15 (𝐾𝑛) ∈ V
64, 5xpex 7465 . . . . . . . . . . . . . 14 ({𝑛} × (𝐾𝑛)) ∈ V
7 axcc2lem.2 . . . . . . . . . . . . . . 15 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
87fvmpt2 6775 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾𝑛)) ∈ V) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
96, 8mpan2 687 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
10 vex 3503 . . . . . . . . . . . . . . 15 𝑛 ∈ V
1110snnz 4710 . . . . . . . . . . . . . 14 {𝑛} ≠ ∅
12 0ex 5208 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
1312snnz 4710 . . . . . . . . . . . . . . . . 17 {∅} ≠ ∅
14 iftrue 4476 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = {∅})
1514neeq1d 3080 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) = ∅ → (if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
1613, 15mpbiri 259 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
17 iffalse 4479 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
18 neqne 3029 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → (𝐹𝑛) ≠ ∅)
1917, 18eqnetrd 3088 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
2016, 19pm2.61i 183 . . . . . . . . . . . . . . 15 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅
21 p0ex 5281 . . . . . . . . . . . . . . . . . 18 {∅} ∈ V
22 fvex 6680 . . . . . . . . . . . . . . . . . 18 (𝐹𝑛) ∈ V
2321, 22ifex 4518 . . . . . . . . . . . . . . . . 17 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V
24 axcc2lem.1 . . . . . . . . . . . . . . . . . 18 𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2524fvmpt2 6775 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2623, 25mpan2 687 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ω → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2726neeq1d 3080 . . . . . . . . . . . . . . 15 (𝑛 ∈ ω → ((𝐾𝑛) ≠ ∅ ↔ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅))
2820, 27mpbiri 259 . . . . . . . . . . . . . 14 (𝑛 ∈ ω → (𝐾𝑛) ≠ ∅)
29 xpnz 6014 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾𝑛)) ≠ ∅)
3029biimpi 217 . . . . . . . . . . . . . 14 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → ({𝑛} × (𝐾𝑛)) ≠ ∅)
3111, 28, 30sylancr 587 . . . . . . . . . . . . 13 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ≠ ∅)
329, 31eqnetrd 3088 . . . . . . . . . . . 12 (𝑛 ∈ ω → (𝐴𝑛) ≠ ∅)
336, 7fnmpti 6488 . . . . . . . . . . . . . 14 𝐴 Fn ω
34 fnfvelrn 6844 . . . . . . . . . . . . . 14 ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴𝑛) ∈ ran 𝐴)
3533, 34mpan 686 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) ∈ ran 𝐴)
36 neeq1 3083 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → (𝑧 ≠ ∅ ↔ (𝐴𝑛) ≠ ∅))
37 fveq2 6667 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → (𝑓𝑧) = (𝑓‘(𝐴𝑛)))
38 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → 𝑧 = (𝐴𝑛))
3937, 38eleq12d 2912 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4036, 39imbi12d 346 . . . . . . . . . . . . . 14 (𝑧 = (𝐴𝑛) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4140rspccv 3624 . . . . . . . . . . . . 13 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ((𝐴𝑛) ∈ ran 𝐴 → ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4235, 41syl5 34 . . . . . . . . . . . 12 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4332, 42mpdi 45 . . . . . . . . . . 11 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4443impcom 408 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))
459eleq2d 2903 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4645adantr 481 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4744, 46mpbid 233 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛)))
48 xp2nd 7713 . . . . . . . . 9 ((𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛)) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
4947, 48syl 17 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
50493adant3 1126 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
512fvmpt2 6775 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
521, 51mpan2 687 . . . . . . . . 9 (𝑛 ∈ ω → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
53523ad2ant1 1127 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
5453eqcomd 2832 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) = (𝐺𝑛))
55263ad2ant1 1127 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
56 ifnefalse 4482 . . . . . . . . 9 ((𝐹𝑛) ≠ ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
57563ad2ant3 1129 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
5855, 57eqtrd 2861 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = (𝐹𝑛))
5950, 54, 583eltr3d 2932 . . . . . 6 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) ∈ (𝐹𝑛))
60593expia 1115 . . . . 5 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
6160expcom 414 . . . 4 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
6261ralrimiv 3186 . . 3 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
63 omex 9095 . . . . 5 ω ∈ V
64 fnex 6975 . . . . 5 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
653, 63, 64mp2an 688 . . . 4 𝐺 ∈ V
66 fneq1 6441 . . . . 5 (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
67 fveq1 6666 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
6867eleq1d 2902 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝐺𝑛) ∈ (𝐹𝑛)))
6968imbi2d 342 . . . . . 6 (𝑔 = 𝐺 → (((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7069ralbidv 3202 . . . . 5 (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7166, 70anbi12d 630 . . . 4 (𝑔 = 𝐺 → ((𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))) ↔ (𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))))
7265, 71spcev 3611 . . 3 ((𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))))
733, 62, 72sylancr 587 . 2 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))))
746a1i 11 . . . . . 6 ((ω ∈ V ∧ 𝑛 ∈ ω) → ({𝑛} × (𝐾𝑛)) ∈ V)
7574, 7fmptd 6874 . . . . 5 (ω ∈ V → 𝐴:ω⟶V)
7663, 75ax-mp 5 . . . 4 𝐴:ω⟶V
77 sneq 4574 . . . . . . . . . 10 (𝑛 = 𝑘 → {𝑛} = {𝑘})
78 fveq2 6667 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐾𝑛) = (𝐾𝑘))
7977, 78xpeq12d 5585 . . . . . . . . 9 (𝑛 = 𝑘 → ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)))
8079, 7, 6fvmpt3i 6770 . . . . . . . 8 (𝑘 ∈ ω → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8180adantl 482 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8281eqeq2d 2837 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) ↔ (𝐴𝑛) = ({𝑘} × (𝐾𝑘))))
839adantr 481 . . . . . . . 8 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
8483eqeq1d 2828 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘))))
85 xp11 6030 . . . . . . . . . 10 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8611, 28, 85sylancr 587 . . . . . . . . 9 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘))))
8710sneqr 4770 . . . . . . . . . 10 ({𝑛} = {𝑘} → 𝑛 = 𝑘)
8887adantr 481 . . . . . . . . 9 (({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘)) → 𝑛 = 𝑘)
8986, 88syl6bi 254 . . . . . . . 8 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9089adantr 481 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9184, 90sylbid 241 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9282, 91sylbid 241 . . . . 5 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘))
9392rgen2 3208 . . . 4 𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)
94 dff13 7007 . . . 4 (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)))
9576, 93, 94mpbir2an 707 . . 3 𝐴:ω–1-1→V
96 f1f1orn 6623 . . . 4 (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
9763f1oen 8519 . . . 4 (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
98 ensym 8547 . . . 4 (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
9996, 97, 983syl 18 . . 3 (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
1007rneqi 5806 . . . . 5 ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
101 dmmptg 6094 . . . . . . . 8 (∀𝑛 ∈ ω ({𝑛} × (𝐾𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω)
1026a1i 11 . . . . . . . 8 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ∈ V)
103101, 102mprg 3157 . . . . . . 7 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω
104103, 63eqeltri 2914 . . . . . 6 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
105 funmpt 6390 . . . . . 6 Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
106 funrnex 7646 . . . . . 6 (dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V → (Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) → ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V))
107104, 105, 106mp2 9 . . . . 5 ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
108100, 107eqeltri 2914 . . . 4 ran 𝐴 ∈ V
109 breq1 5066 . . . . 5 (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
110 raleq 3411 . . . . . 6 (𝑎 = ran 𝐴 → (∀𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
111110exbidv 1915 . . . . 5 (𝑎 = ran 𝐴 → (∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
112109, 111imbi12d 346 . . . 4 (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
113 ax-cc 9846 . . . 4 (𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
114108, 112, 113vtocl 3565 . . 3 (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
11595, 99, 114mp2b 10 . 2 𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
11673, 115exlimiiv 1925 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  Vcvv 3500  c0 4295  ifcif 4470  {csn 4564   class class class wbr 5063  cmpt 5143   × cxp 5552  dom cdm 5554  ran crn 5555  Fun wfun 6346   Fn wfn 6347  wf 6348  1-1wf1 6349  1-1-ontowf1o 6351  cfv 6352  ωcom 7568  2nd c2nd 7679  cen 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cc 9846
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-2nd 7681  df-er 8279  df-en 8499
This theorem is referenced by:  axcc2  9848
  Copyright terms: Public domain W3C validator