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Theorem isf34lem6 9240
Description: Lemma for isfin3-4 9242. (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 7921 . . . 4 (𝑓 ∈ (𝒫 𝐴𝑚 ω) → 𝑓:ω⟶𝒫 𝐴)
2 compss.a . . . . . 6 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32isf34lem7 9239 . . . . 5 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦)) → ran 𝑓 ∈ ran 𝑓)
433expia 1286 . . . 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 2995 . 2 (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
7 elmapex 7920 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
87simpld 474 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝒫 𝐴 ∈ V)
9 pwexb 7017 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 224 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐴 ∈ V)
112isf34lem2 9233 . . . . . . . . 9 (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13 elmapi 7921 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝑔:ω⟶𝒫 𝐴)
14 fco 6096 . . . . . . . 8 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝑔:ω⟶𝒫 𝐴) → (𝐹𝑔):ω⟶𝒫 𝐴)
1512, 13, 14syl2anc 694 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔):ω⟶𝒫 𝐴)
16 elmapg 7912 . . . . . . . 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 247 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔) ∈ (𝒫 𝐴𝑚 ω))
19 fveq1 6228 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓𝑦) = ((𝐹𝑔)‘𝑦))
20 fveq1 6228 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘suc 𝑦) = ((𝐹𝑔)‘suc 𝑦))
2119, 20sseq12d 3667 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
2221ralbidv 3015 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
23 rneq 5383 . . . . . . . . . . 11 (𝑓 = (𝐹𝑔) → ran 𝑓 = ran (𝐹𝑔))
24 rnco2 5680 . . . . . . . . . . 11 ran (𝐹𝑔) = (𝐹 “ ran 𝑔)
2523, 24syl6eq 2701 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2625unieqd 4478 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2726, 25eleq12d 2724 . . . . . . . 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 3337 . . . . . 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 34 . . . . 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 3777 . . . . . . . . 9 ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
3210adantr 480 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
3313ffvelrnda 6399 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ∈ 𝒫 𝐴)
3433elpwid 4203 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ⊆ 𝐴)
352isf34lem1 9232 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔𝑦) ⊆ 𝐴) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
3632, 34, 35syl2anc 694 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
37 peano2 7128 . . . . . . . . . . . . 13 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
38 ffvelrn 6397 . . . . . . . . . . . . 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 4203 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
412isf34lem1 9232 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4232, 40, 41syl2anc 694 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4336, 42sseq12d 3667 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
4431, 43syl5ibr 236 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
45 fvco3 6314 . . . . . . . . . 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 6314 . . . . . . . . . 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 3667 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
5044, 49sylibrd 249 . . . . . . 7 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5150ralimdva 2991 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
52 ffn 6083 . . . . . . . . 9 (𝐹:𝒫 𝐴⟶𝒫 𝐴𝐹 Fn 𝒫 𝐴)
5312, 52syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹 Fn 𝒫 𝐴)
54 imassrn 5512 . . . . . . . . 9 (𝐹 “ ran 𝑔) ⊆ ran 𝐹
55 frn 6091 . . . . . . . . . 10 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴)
5612, 55syl 17 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝐹 ⊆ 𝒫 𝐴)
5754, 56syl5ss 3647 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
58 fnfvima 6536 . . . . . . . . 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 1286 . . . . . . . 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 694 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
61 incom 3838 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
62 frn 6091 . . . . . . . . . . . . . . . 16 (𝑔:ω⟶𝒫 𝐴 → ran 𝑔 ⊆ 𝒫 𝐴)
6313, 62syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ 𝒫 𝐴)
64 fdm 6089 . . . . . . . . . . . . . . . 16 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → dom 𝐹 = 𝒫 𝐴)
6512, 64syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝐹 = 𝒫 𝐴)
6663, 65sseqtr4d 3675 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ dom 𝐹)
67 df-ss 3621 . . . . . . . . . . . . . 14 (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6866, 67sylib 208 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6961, 68syl5eq 2697 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
70 fdm 6089 . . . . . . . . . . . . . . 15 (𝑔:ω⟶𝒫 𝐴 → dom 𝑔 = ω)
7113, 70syl 17 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 = ω)
72 peano1 7127 . . . . . . . . . . . . . . 15 ∅ ∈ ω
73 ne0i 3954 . . . . . . . . . . . . . . 15 (∅ ∈ ω → ω ≠ ∅)
7472, 73mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ω ≠ ∅)
7571, 74eqnetrd 2890 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 ≠ ∅)
76 dm0rn0 5374 . . . . . . . . . . . . . 14 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
7776necon3bii 2875 . . . . . . . . . . . . 13 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
7875, 77sylib 208 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ≠ ∅)
7969, 78eqnetrd 2890 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
80 imadisj 5519 . . . . . . . . . . . 12 ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
8180necon3bii 2875 . . . . . . . . . . 11 ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
8279, 81sylibr 224 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ≠ ∅)
832isf34lem4 9237 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
8410, 57, 82, 83syl12anc 1364 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
852isf34lem3 9235 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8610, 63, 85syl2anc 694 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8786inteqd 4512 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8884, 87eqtrd 2685 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = ran 𝑔)
8988, 86eleq12d 2724 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ((𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ran 𝑔 ∈ ran 𝑔))
9060, 89sylibd 229 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → ran 𝑔 ∈ ran 𝑔))
9151, 90imim12d 81 . . . . 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 2994 . . 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 9228 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
9593, 94syl5ibr 236 . 2 (𝐴𝑉 → (∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
966, 95impbid2 216 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 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   cint 4507  cmpt 4762  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  suc csuc 5763   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  𝑚 cmap 7899  FinIIIcfin3 9141
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
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-rmo 2949  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-int 4508  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-se 5103  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-pred 5718  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-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-rpss 6979  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-wdom 8505  df-card 8803  df-fin4 9147  df-fin3 9148
This theorem is referenced by:  isfin3-4  9242
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