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Theorem axcc2lem 10474
Description: Lemma for axcc2 10475. (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 6920 . . . 4 (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V
2 axcc2lem.3 . . . 4 𝐺 = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘(𝐴𝑛))))
31, 2fnmpti 6712 . . 3 𝐺 Fn ω
4 vsnex 5440 . . . . . . . . . . . . . . 15 {𝑛} ∈ V
5 fvex 6920 . . . . . . . . . . . . . . 15 (𝐾𝑛) ∈ V
64, 5xpex 7772 . . . . . . . . . . . . . 14 ({𝑛} × (𝐾𝑛)) ∈ V
7 axcc2lem.2 . . . . . . . . . . . . . . 15 𝐴 = (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
87fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ ({𝑛} × (𝐾𝑛)) ∈ V) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
96, 8mpan2 691 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
10 vex 3482 . . . . . . . . . . . . . . 15 𝑛 ∈ V
1110snnz 4781 . . . . . . . . . . . . . 14 {𝑛} ≠ ∅
12 0ex 5313 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
1312snnz 4781 . . . . . . . . . . . . . . . . 17 {∅} ≠ ∅
14 iftrue 4537 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = {∅})
1514neeq1d 2998 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) = ∅ → (if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅ ↔ {∅} ≠ ∅))
1613, 15mpbiri 258 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
17 iffalse 4540 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
18 neqne 2946 . . . . . . . . . . . . . . . . 17 (¬ (𝐹𝑛) = ∅ → (𝐹𝑛) ≠ ∅)
1917, 18eqnetrd 3006 . . . . . . . . . . . . . . . 16 (¬ (𝐹𝑛) = ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅)
2016, 19pm2.61i 182 . . . . . . . . . . . . . . 15 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅
21 p0ex 5390 . . . . . . . . . . . . . . . . . 18 {∅} ∈ V
22 fvex 6920 . . . . . . . . . . . . . . . . . 18 (𝐹𝑛) ∈ V
2321, 22ifex 4581 . . . . . . . . . . . . . . . . 17 if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V
24 axcc2lem.1 . . . . . . . . . . . . . . . . . 18 𝐾 = (𝑛 ∈ ω ↦ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2524fvmpt2 7027 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ω ∧ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ∈ V) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2623, 25mpan2 691 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ω → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
2726neeq1d 2998 . . . . . . . . . . . . . . 15 (𝑛 ∈ ω → ((𝐾𝑛) ≠ ∅ ↔ if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) ≠ ∅))
2820, 27mpbiri 258 . . . . . . . . . . . . . 14 (𝑛 ∈ ω → (𝐾𝑛) ≠ ∅)
29 xpnz 6181 . . . . . . . . . . . . . . 15 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) ↔ ({𝑛} × (𝐾𝑛)) ≠ ∅)
3029biimpi 216 . . . . . . . . . . . . . 14 (({𝑛} ≠ ∅ ∧ (𝐾𝑛) ≠ ∅) → ({𝑛} × (𝐾𝑛)) ≠ ∅)
3111, 28, 30sylancr 587 . . . . . . . . . . . . 13 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ≠ ∅)
329, 31eqnetrd 3006 . . . . . . . . . . . 12 (𝑛 ∈ ω → (𝐴𝑛) ≠ ∅)
336, 7fnmpti 6712 . . . . . . . . . . . . . 14 𝐴 Fn ω
34 fnfvelrn 7100 . . . . . . . . . . . . . 14 ((𝐴 Fn ω ∧ 𝑛 ∈ ω) → (𝐴𝑛) ∈ ran 𝐴)
3533, 34mpan 690 . . . . . . . . . . . . 13 (𝑛 ∈ ω → (𝐴𝑛) ∈ ran 𝐴)
36 neeq1 3001 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → (𝑧 ≠ ∅ ↔ (𝐴𝑛) ≠ ∅))
37 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → (𝑓𝑧) = (𝑓‘(𝐴𝑛)))
38 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐴𝑛) → 𝑧 = (𝐴𝑛))
3937, 38eleq12d 2833 . . . . . . . . . . . . . . 15 (𝑧 = (𝐴𝑛) → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛)))
4036, 39imbi12d 344 . . . . . . . . . . . . . 14 (𝑧 = (𝐴𝑛) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ((𝐴𝑛) ≠ ∅ → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))))
4140rspccv 3619 . . . . . . . . . . . . 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 407 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛))
459eleq2d 2825 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4645adantr 480 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝑓‘(𝐴𝑛)) ∈ (𝐴𝑛) ↔ (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛))))
4744, 46mpbid 232 . . . . . . . . 9 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑓‘(𝐴𝑛)) ∈ ({𝑛} × (𝐾𝑛)))
48 xp2nd 8046 . . . . . . . . 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 1131 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) ∈ (𝐾𝑛))
512fvmpt2 7027 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ (2nd ‘(𝑓‘(𝐴𝑛))) ∈ V) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
521, 51mpan2 691 . . . . . . . . 9 (𝑛 ∈ ω → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
53523ad2ant1 1132 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) = (2nd ‘(𝑓‘(𝐴𝑛))))
5453eqcomd 2741 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (2nd ‘(𝑓‘(𝐴𝑛))) = (𝐺𝑛))
55263ad2ant1 1132 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)))
56 ifnefalse 4543 . . . . . . . . 9 ((𝐹𝑛) ≠ ∅ → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
57563ad2ant3 1134 . . . . . . . 8 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → if((𝐹𝑛) = ∅, {∅}, (𝐹𝑛)) = (𝐹𝑛))
5855, 57eqtrd 2775 . . . . . . 7 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐾𝑛) = (𝐹𝑛))
5950, 54, 583eltr3d 2853 . . . . . 6 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ∧ (𝐹𝑛) ≠ ∅) → (𝐺𝑛) ∈ (𝐹𝑛))
60593expia 1120 . . . . 5 ((𝑛 ∈ ω ∧ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
6160expcom 413 . . . 4 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → (𝑛 ∈ ω → ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
6261ralrimiv 3143 . . 3 (∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))
63 omex 9681 . . . . 5 ω ∈ V
64 fnex 7237 . . . . 5 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
653, 63, 64mp2an 692 . . . 4 𝐺 ∈ V
66 fneq1 6660 . . . . 5 (𝑔 = 𝐺 → (𝑔 Fn ω ↔ 𝐺 Fn ω))
67 fveq1 6906 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑛) = (𝐺𝑛))
6867eleq1d 2824 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔𝑛) ∈ (𝐹𝑛) ↔ (𝐺𝑛) ∈ (𝐹𝑛)))
6968imbi2d 340 . . . . . 6 (𝑔 = 𝐺 → (((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7069ralbidv 3176 . . . . 5 (𝑔 = 𝐺 → (∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)) ↔ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛))))
7166, 70anbi12d 632 . . . 4 (𝑔 = 𝐺 → ((𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛))) ↔ (𝐺 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝐺𝑛) ∈ (𝐹𝑛)))))
7265, 71spcev 3606 . . 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 7134 . . . . 5 (ω ∈ V → 𝐴:ω⟶V)
7663, 75ax-mp 5 . . . 4 𝐴:ω⟶V
77 sneq 4641 . . . . . . . . . 10 (𝑛 = 𝑘 → {𝑛} = {𝑘})
78 fveq2 6907 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝐾𝑛) = (𝐾𝑘))
7977, 78xpeq12d 5720 . . . . . . . . 9 (𝑛 = 𝑘 → ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)))
8079, 7, 6fvmpt3i 7021 . . . . . . . 8 (𝑘 ∈ ω → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8180adantl 481 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑘) = ({𝑘} × (𝐾𝑘)))
8281eqeq2d 2746 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) ↔ (𝐴𝑛) = ({𝑘} × (𝐾𝑘))))
839adantr 480 . . . . . . . 8 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (𝐴𝑛) = ({𝑛} × (𝐾𝑛)))
8483eqeq1d 2737 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) ↔ ({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘))))
85 xp11 6197 . . . . . . . . . 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 4845 . . . . . . . . . 10 ({𝑛} = {𝑘} → 𝑛 = 𝑘)
8887adantr 480 . . . . . . . . 9 (({𝑛} = {𝑘} ∧ (𝐾𝑛) = (𝐾𝑘)) → 𝑛 = 𝑘)
8986, 88biimtrdi 253 . . . . . . . 8 (𝑛 ∈ ω → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9089adantr 480 . . . . . . 7 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → (({𝑛} × (𝐾𝑛)) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9184, 90sylbid 240 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = ({𝑘} × (𝐾𝑘)) → 𝑛 = 𝑘))
9282, 91sylbid 240 . . . . 5 ((𝑛 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘))
9392rgen2 3197 . . . 4 𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)
94 dff13 7275 . . . 4 (𝐴:ω–1-1→V ↔ (𝐴:ω⟶V ∧ ∀𝑛 ∈ ω ∀𝑘 ∈ ω ((𝐴𝑛) = (𝐴𝑘) → 𝑛 = 𝑘)))
9576, 93, 94mpbir2an 711 . . 3 𝐴:ω–1-1→V
96 f1f1orn 6860 . . . 4 (𝐴:ω–1-1→V → 𝐴:ω–1-1-onto→ran 𝐴)
9763f1oen 9012 . . . 4 (𝐴:ω–1-1-onto→ran 𝐴 → ω ≈ ran 𝐴)
98 ensym 9042 . . . 4 (ω ≈ ran 𝐴 → ran 𝐴 ≈ ω)
9996, 97, 983syl 18 . . 3 (𝐴:ω–1-1→V → ran 𝐴 ≈ ω)
1007rneqi 5951 . . . . 5 ran 𝐴 = ran (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
101 dmmptg 6264 . . . . . . . 8 (∀𝑛 ∈ ω ({𝑛} × (𝐾𝑛)) ∈ V → dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω)
1026a1i 11 . . . . . . . 8 (𝑛 ∈ ω → ({𝑛} × (𝐾𝑛)) ∈ V)
103101, 102mprg 3065 . . . . . . 7 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) = ω
104103, 63eqeltri 2835 . . . . . 6 dom (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛))) ∈ V
105 funmpt 6606 . . . . . 6 Fun (𝑛 ∈ ω ↦ ({𝑛} × (𝐾𝑛)))
106 funrnex 7977 . . . . . 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 2835 . . . 4 ran 𝐴 ∈ V
109 breq1 5151 . . . . 5 (𝑎 = ran 𝐴 → (𝑎 ≈ ω ↔ ran 𝐴 ≈ ω))
110 raleq 3321 . . . . . 6 (𝑎 = ran 𝐴 → (∀𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
111110exbidv 1919 . . . . 5 (𝑎 = ran 𝐴 → (∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
112109, 111imbi12d 344 . . . 4 (𝑎 = ran 𝐴 → ((𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
113 ax-cc 10473 . . . 4 (𝑎 ≈ ω → ∃𝑓𝑧𝑎 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
114108, 112, 113vtocl 3558 . . 3 (ran 𝐴 ≈ ω → ∃𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
11595, 99, 114mp2b 10 . 2 𝑓𝑧 ∈ ran 𝐴(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)
11673, 115exlimiiv 1929 1 𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω ((𝐹𝑛) ≠ ∅ → (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  Vcvv 3478  c0 4339  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  ωcom 7887  2nd c2nd 8012  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-2nd 8014  df-er 8744  df-en 8985
This theorem is referenced by:  axcc2  10475
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