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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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