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Theorem isf34lem6 9796
Description: Lemma for isfin3-4 9798. (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 8423 . . . 4 (𝑓 ∈ (𝒫 𝐴m ω) → 𝑓:ω⟶𝒫 𝐴)
2 compss.a . . . . . 6 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32isf34lem7 9795 . . . . 5 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦)) → ran 𝑓 ∈ ran 𝑓)
433expia 1115 . . . 4 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
51, 4sylan2 592 . . 3 ((𝐴 ∈ FinIII𝑓 ∈ (𝒫 𝐴m ω)) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
65ralrimiva 3187 . 2 (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
7 elmapex 8422 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴m ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
87simpld 495 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴m ω) → 𝒫 𝐴 ∈ V)
9 pwexb 7481 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 235 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → 𝐴 ∈ V)
112isf34lem2 9789 . . . . . . . . 9 (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13 elmapi 8423 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → 𝑔:ω⟶𝒫 𝐴)
14 fco 6530 . . . . . . . 8 ((𝐹:𝒫 𝐴⟶𝒫 𝐴𝑔:ω⟶𝒫 𝐴) → (𝐹𝑔):ω⟶𝒫 𝐴)
1512, 13, 14syl2anc 584 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹𝑔):ω⟶𝒫 𝐴)
16 elmapg 8414 . . . . . . . 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 258 . . . . . 6 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹𝑔) ∈ (𝒫 𝐴m ω))
19 fveq1 6668 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓𝑦) = ((𝐹𝑔)‘𝑦))
20 fveq1 6668 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘suc 𝑦) = ((𝐹𝑔)‘suc 𝑦))
2119, 20sseq12d 4004 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
2221ralbidv 3202 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
23 rneq 5805 . . . . . . . . . . 11 (𝑓 = (𝐹𝑔) → ran 𝑓 = ran (𝐹𝑔))
24 rnco2 6105 . . . . . . . . . . 11 ran (𝐹𝑔) = (𝐹 “ ran 𝑔)
2523, 24syl6eq 2877 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2625unieqd 4847 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2726, 25eleq12d 2912 . . . . . . . 8 (𝑓 = (𝐹𝑔) → ( ran 𝑓 ∈ ran 𝑓 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
2822, 27imbi12d 346 . . . . . . 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 3624 . . . . . 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 4119 . . . . . . . . 9 ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
3213ffvelrnda 6849 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ∈ 𝒫 𝐴)
3332elpwid 4556 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ⊆ 𝐴)
342isf34lem1 9788 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔𝑦) ⊆ 𝐴) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
3510, 33, 34syl2an2r 681 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
36 peano2 7595 . . . . . . . . . . . . 13 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
37 ffvelrn 6847 . . . . . . . . . . . . 13 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3813, 36, 37syl2an 595 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3938elpwid 4556 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
402isf34lem1 9788 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4110, 39, 40syl2an2r 681 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4235, 41sseq12d 4004 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
4331, 42syl5ibr 247 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
44 fvco3 6759 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
4513, 44sylan 580 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘𝑦) = (𝐹‘(𝑔𝑦)))
46 fvco3 6759 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4713, 36, 46syl2an 595 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4845, 47sseq12d 4004 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
4943, 48sylibrd 260 . . . . . . 7 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5049ralimdva 3182 . . . . . 6 (𝑔 ∈ (𝒫 𝐴m ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5112ffnd 6514 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → 𝐹 Fn 𝒫 𝐴)
52 imassrn 5939 . . . . . . . . 9 (𝐹 “ ran 𝑔) ⊆ ran 𝐹
5312frnd 6520 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝐹 ⊆ 𝒫 𝐴)
5452, 53sstrid 3982 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
55 fnfvima 6991 . . . . . . . . 9 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)))
56553expia 1115 . . . . . . . 8 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
5751, 54, 56syl2anc 584 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴m ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → (𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔))))
58 incom 4182 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
5913frnd 6520 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝑔 ⊆ 𝒫 𝐴)
6012fdmd 6522 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴m ω) → dom 𝐹 = 𝒫 𝐴)
6159, 60sseqtrrd 4012 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝑔 ⊆ dom 𝐹)
62 df-ss 3956 . . . . . . . . . . . . . 14 (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6361, 62sylib 219 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴m ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6458, 63syl5eq 2873 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴m ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
6513fdmd 6522 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴m ω) → dom 𝑔 = ω)
66 peano1 7594 . . . . . . . . . . . . . . 15 ∅ ∈ ω
67 ne0i 4304 . . . . . . . . . . . . . . 15 (∅ ∈ ω → ω ≠ ∅)
6866, 67mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴m ω) → ω ≠ ∅)
6965, 68eqnetrd 3088 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴m ω) → dom 𝑔 ≠ ∅)
70 dm0rn0 5794 . . . . . . . . . . . . . 14 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
7170necon3bii 3073 . . . . . . . . . . . . 13 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
7269, 71sylib 219 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝑔 ≠ ∅)
7364, 72eqnetrd 3088 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴m ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
74 imadisj 5947 . . . . . . . . . . . 12 ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
7574necon3bii 3073 . . . . . . . . . . 11 ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
7673, 75sylibr 235 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ ran 𝑔) ≠ ∅)
772isf34lem4 9793 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
7810, 54, 76, 77syl12anc 834 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
792isf34lem3 9791 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ran 𝑔 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8010, 59, 79syl2anc 584 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8180inteqd 4879 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8278, 81eqtrd 2861 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 (𝐹 “ ran 𝑔)) = ran 𝑔)
8382, 80eleq12d 2912 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴m ω) → ((𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ran 𝑔 ∈ ran 𝑔))
8457, 83sylibd 240 . . . . . 6 (𝑔 ∈ (𝒫 𝐴m ω) → ( (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔) → ran 𝑔 ∈ ran 𝑔))
8550, 84imim12d 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)))
8630, 85sylcom 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)))
8786ralrimiv 3186 . . 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))
88 isfin3-3 9784 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
8987, 88syl5ibr 247 . 2 (𝐴𝑉 → (∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
906, 89impbid2 227 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 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  Vcvv 3500  cdif 3937  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542   cuni 4837   cint 4874  cmpt 5143  dom cdm 5554  ran crn 5555  cima 5557  ccom 5558  suc csuc 6192   Fn wfn 6349  wf 6350  cfv 6354  (class class class)co 7150  ωcom 7573  m cmap 8401  FinIIIcfin3 9697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rpss 7443  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seqom 8080  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-wdom 9017  df-card 9362  df-fin4 9703  df-fin3 9704
This theorem is referenced by:  isfin3-4  9798
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