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Theorem isf34lem6 9085
 Description: Lemma for isfin3-4 9087. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
Assertion
Ref Expression
isf34lem6 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓)))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝑓,𝐹,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑓)

Proof of Theorem isf34lem6
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elmapi 7765 . . . 4 (𝑓 ∈ (𝒫 𝐴𝑚 ω) → 𝑓:ω⟶𝒫 𝐴)
2 compss.a . . . . . 6 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32isf34lem7 9084 . . . . 5 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦)) → ran 𝑓 ∈ ran 𝑓)
433expia 1259 . . . 4 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
51, 4sylan2 490 . . 3 ((𝐴 ∈ FinIII𝑓 ∈ (𝒫 𝐴𝑚 ω)) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
65ralrimiva 2949 . 2 (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
7 elmapex 7764 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
87simpld 474 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝒫 𝐴 ∈ V)
9 pwexb 6867 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 223 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐴 ∈ V)
112isf34lem2 9078 . . . . . . . . 9 (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13 elmapi 7765 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝑔:ω⟶𝒫 𝐴)
14 fco 5971 . . . . . . . 8 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝑔:ω⟶𝒫 𝐴) → (𝐹𝑔):ω⟶𝒫 𝐴)
1512, 13, 14syl2anc 691 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔):ω⟶𝒫 𝐴)
16 elmapg 7757 . . . . . . . 8 ((𝒫 𝐴 ∈ V ∧ ω ∈ V) → ((𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω) ↔ (𝐹𝑔):ω⟶𝒫 𝐴))
177, 16syl 17 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω) ↔ (𝐹𝑔):ω⟶𝒫 𝐴))
1815, 17mpbird 246 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω))
19 fveq1 6102 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓𝑦) = ((𝐹𝑔)‘𝑦))
20 fveq1 6102 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘suc 𝑦) = ((𝐹𝑔)‘suc 𝑦))
2119, 20sseq12d 3597 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
2221ralbidv 2969 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
23 rneq 5272 . . . . . . . . . . 11 (𝑓 = (𝐹𝑔) → ran 𝑓 = ran (𝐹𝑔))
24 rnco2 5559 . . . . . . . . . . 11 ran (𝐹𝑔) = (𝐹 “ ran 𝑔)
2523, 24syl6eq 2660 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2625unieqd 4382 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2726, 25eleq12d 2682 . . . . . . . 8 (𝑓 = (𝐹𝑔) → ( ran 𝑓 ∈ ran 𝑓 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
2822, 27imbi12d 333 . . . . . . 7 (𝑓 = (𝐹𝑔) → ((∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) ↔ (∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
2928rspccv 3279 . . . . . 6 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → ((𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
3018, 29syl5 33 . . . . 5 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔))))
31 sscon 3706 . . . . . . . . 9 ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
3210adantr 480 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
3313ffvelrnda 6267 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ∈ 𝒫 𝐴)
3433elpwid 4118 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ⊆ 𝐴)
352isf34lem1 9077 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔𝑦) ⊆ 𝐴) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
3632, 34, 35syl2anc 691 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
37 peano2 6978 . . . . . . . . . . . . 13 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
38 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3913, 37, 38syl2an 493 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
4039elpwid 4118 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
412isf34lem1 9077 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4232, 40, 41syl2anc 691 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4336, 42sseq12d 3597 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
4431, 43syl5ibr 235 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
45 fvco3 6185 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
4613, 45sylan 487 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
47 fvco3 6185 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4813, 37, 47syl2an 493 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4946, 48sseq12d 3597 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
5044, 49sylibrd 248 . . . . . . 7 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5150ralimdva 2945 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
52 ffn 5958 . . . . . . . . 9 (𝐹:𝒫 𝐴⟶𝒫 𝐴𝐹 Fn 𝒫 𝐴)
5312, 52syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹 Fn 𝒫 𝐴)
54 imassrn 5396 . . . . . . . . 9 (𝐹 “ ran 𝑔) ⊆ ran 𝐹
55 frn 5966 . . . . . . . . . 10 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴)
5612, 55syl 17 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝐹 ⊆ 𝒫 𝐴)
5754, 56syl5ss 3579 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
58 fnfvima 6400 . . . . . . . . 9 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)))
59583expia 1259 . . . . . . . 8 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
6053, 57, 59syl2anc 691 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
61 incom 3767 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
62 frn 5966 . . . . . . . . . . . . . . . 16 (𝑔:ω⟶𝒫 𝐴 → ran 𝑔 ⊆ 𝒫 𝐴)
6313, 62syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ 𝒫 𝐴)
64 fdm 5964 . . . . . . . . . . . . . . . 16 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → dom 𝐹 = 𝒫 𝐴)
6512, 64syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝐹 = 𝒫 𝐴)
6663, 65sseqtr4d 3605 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ dom 𝐹)
67 df-ss 3554 . . . . . . . . . . . . . 14 (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6866, 67sylib 207 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6961, 68syl5eq 2656 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
70 fdm 5964 . . . . . . . . . . . . . . 15 (𝑔:ω⟶𝒫 𝐴 → dom 𝑔 = ω)
7113, 70syl 17 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 = ω)
72 peano1 6977 . . . . . . . . . . . . . . 15 ∅ ∈ ω
73 ne0i 3880 . . . . . . . . . . . . . . 15 (∅ ∈ ω → ω ≠ ∅)
7472, 73mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ω ≠ ∅)
7571, 74eqnetrd 2849 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 ≠ ∅)
76 dm0rn0 5263 . . . . . . . . . . . . . 14 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
7776necon3bii 2834 . . . . . . . . . . . . 13 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
7875, 77sylib 207 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ≠ ∅)
7969, 78eqnetrd 2849 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
80 imadisj 5403 . . . . . . . . . . . 12 ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
8180necon3bii 2834 . . . . . . . . . . 11 ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
8279, 81sylibr 223 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ≠ ∅)
832isf34lem4 9082 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
8410, 57, 82, 83syl12anc 1316 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
852isf34lem3 9080 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8610, 63, 85syl2anc 691 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8786inteqd 4415 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8884, 87eqtrd 2644 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = ran 𝑔)
8988, 86eleq12d 2682 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ran 𝑔 ∈ ran 𝑔))
9060, 89sylibd 228 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → ran 𝑔 ∈ ran 𝑔))
9151, 90imim12d 79 . . . . 5 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) → (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
9230, 91sylcom 30 . . . 4 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
9392ralrimiv 2948 . . 3 (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → ∀𝑔 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔))
94 isfin3-3 9073 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
9593, 94syl5ibr 235 . 2 (𝐴𝑉 → (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
966, 95impbid2 215 1 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ↑𝑚 cmap 7744  FinIIIcfin3 8986 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rpss 6835  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wdom 8347  df-card 8648  df-fin4 8992  df-fin3 8993 This theorem is referenced by:  isfin3-4  9087
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