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Theorem isf34lem6 9146
Description: Lemma for isfin3-4 9148. (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 7823 . . . 4 (𝑓 ∈ (𝒫 𝐴𝑚 ω) → 𝑓:ω⟶𝒫 𝐴)
2 compss.a . . . . . 6 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32isf34lem7 9145 . . . . 5 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦)) → ran 𝑓 ∈ ran 𝑓)
433expia 1264 . . . 4 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
51, 4sylan2 491 . . 3 ((𝐴 ∈ FinIII𝑓 ∈ (𝒫 𝐴𝑚 ω)) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
65ralrimiva 2960 . 2 (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
7 elmapex 7822 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
87simpld 475 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝒫 𝐴 ∈ V)
9 pwexb 6922 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 224 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐴 ∈ V)
112isf34lem2 9139 . . . . . . . . 9 (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13 elmapi 7823 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝑔:ω⟶𝒫 𝐴)
14 fco 6015 . . . . . . . 8 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝑔:ω⟶𝒫 𝐴) → (𝐹𝑔):ω⟶𝒫 𝐴)
1512, 13, 14syl2anc 692 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹𝑔):ω⟶𝒫 𝐴)
16 elmapg 7815 . . . . . . . 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 6147 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓𝑦) = ((𝐹𝑔)‘𝑦))
20 fveq1 6147 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘suc 𝑦) = ((𝐹𝑔)‘suc 𝑦))
2119, 20sseq12d 3613 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
2221ralbidv 2980 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
23 rneq 5311 . . . . . . . . . . 11 (𝑓 = (𝐹𝑔) → ran 𝑓 = ran (𝐹𝑔))
24 rnco2 5601 . . . . . . . . . . 11 ran (𝐹𝑔) = (𝐹 “ ran 𝑔)
2523, 24syl6eq 2671 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2625unieqd 4412 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2726, 25eleq12d 2692 . . . . . . . 8 (𝑓 = (𝐹𝑔) → ( ran 𝑓 ∈ ran 𝑓 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
2822, 27imbi12d 334 . . . . . . 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 3292 . . . . . 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 3722 . . . . . . . . 9 ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
3210adantr 481 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → 𝐴 ∈ V)
3313ffvelrnda 6315 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ∈ 𝒫 𝐴)
3433elpwid 4141 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ⊆ 𝐴)
352isf34lem1 9138 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔𝑦) ⊆ 𝐴) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
3632, 34, 35syl2anc 692 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
37 peano2 7033 . . . . . . . . . . . . 13 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
38 ffvelrn 6313 . . . . . . . . . . . . 13 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3913, 37, 38syl2an 494 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
4039elpwid 4141 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
412isf34lem1 9138 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4232, 40, 41syl2anc 692 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4336, 42sseq12d 3613 . . . . . . . . 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 6232 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
4613, 45sylan 488 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
47 fvco3 6232 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4813, 37, 47syl2an 494 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴𝑚 ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4946, 48sseq12d 3613 . . . . . . . 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 2956 . . . . . 6 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
52 ffn 6002 . . . . . . . . 9 (𝐹:𝒫 𝐴⟶𝒫 𝐴𝐹 Fn 𝒫 𝐴)
5312, 52syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → 𝐹 Fn 𝒫 𝐴)
54 imassrn 5436 . . . . . . . . 9 (𝐹 “ ran 𝑔) ⊆ ran 𝐹
55 frn 6010 . . . . . . . . . 10 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴)
5612, 55syl 17 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝐹 ⊆ 𝒫 𝐴)
5754, 56syl5ss 3594 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
58 fnfvima 6450 . . . . . . . . 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 1264 . . . . . . . 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 692 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
61 incom 3783 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
62 frn 6010 . . . . . . . . . . . . . . . 16 (𝑔:ω⟶𝒫 𝐴 → ran 𝑔 ⊆ 𝒫 𝐴)
6313, 62syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ 𝒫 𝐴)
64 fdm 6008 . . . . . . . . . . . . . . . 16 (𝐹:𝒫 𝐴⟶𝒫 𝐴 → dom 𝐹 = 𝒫 𝐴)
6512, 64syl 17 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝐹 = 𝒫 𝐴)
6663, 65sseqtr4d 3621 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ⊆ dom 𝐹)
67 df-ss 3569 . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
70 fdm 6008 . . . . . . . . . . . . . . 15 (𝑔:ω⟶𝒫 𝐴 → dom 𝑔 = ω)
7113, 70syl 17 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 = ω)
72 peano1 7032 . . . . . . . . . . . . . . 15 ∅ ∈ ω
73 ne0i 3897 . . . . . . . . . . . . . . 15 (∅ ∈ ω → ω ≠ ∅)
7472, 73mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ω ≠ ∅)
7571, 74eqnetrd 2857 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → dom 𝑔 ≠ ∅)
76 dm0rn0 5302 . . . . . . . . . . . . . 14 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
7776necon3bii 2842 . . . . . . . . . . . . 13 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
7875, 77sylib 208 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → ran 𝑔 ≠ ∅)
7969, 78eqnetrd 2857 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
80 imadisj 5443 . . . . . . . . . . . 12 ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
8180necon3bii 2842 . . . . . . . . . . 11 ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
8279, 81sylibr 224 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ ran 𝑔) ≠ ∅)
832isf34lem4 9143 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
8410, 57, 82, 83syl12anc 1321 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
852isf34lem3 9141 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8610, 63, 85syl2anc 692 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8786inteqd 4445 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8884, 87eqtrd 2655 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴𝑚 ω) → (𝐹 (𝐹 “ ran 𝑔)) = ran 𝑔)
8988, 86eleq12d 2692 . . . . . . 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 2959 . . 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 9134 . . 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 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   cint 4440  cmpt 4673  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  suc csuc 5684   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  ωcom 7012  𝑚 cmap 7802  FinIIIcfin3 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-rpss 6890  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wdom 8408  df-card 8709  df-fin4 9053  df-fin3 9054
This theorem is referenced by:  isfin3-4  9148
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