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Theorem isf34lem6 10417
Description: Lemma for isfin3-4 10419. (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 8887 . . . 4 (𝑓 ∈ (𝒫 𝐴m ω) → 𝑓:ω⟶𝒫 𝐴)
2 compss.a . . . . . 6 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32isf34lem7 10416 . . . . 5 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦)) → ran 𝑓 ∈ ran 𝑓)
433expia 1120 . . . 4 ((𝐴 ∈ FinIII𝑓:ω⟶𝒫 𝐴) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
51, 4sylan2 593 . . 3 ((𝐴 ∈ FinIII𝑓 ∈ (𝒫 𝐴m ω)) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
65ralrimiva 3143 . 2 (𝐴 ∈ FinIII → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓))
7 elmapex 8886 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴m ω) → (𝒫 𝐴 ∈ V ∧ ω ∈ V))
87simpld 494 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴m ω) → 𝒫 𝐴 ∈ V)
9 pwexb 7784 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 234 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → 𝐴 ∈ V)
112isf34lem2 10410 . . . . . . . . 9 (𝐴 ∈ V → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
13 elmapi 8887 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → 𝑔:ω⟶𝒫 𝐴)
14 fco 6760 . . . . . . . 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 8877 . . . . . . . 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 257 . . . . . 6 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹𝑔) ∈ (𝒫 𝐴m ω))
19 fveq1 6905 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓𝑦) = ((𝐹𝑔)‘𝑦))
20 fveq1 6905 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘suc 𝑦) = ((𝐹𝑔)‘suc 𝑦))
2119, 20sseq12d 4028 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
2221ralbidv 3175 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) ↔ ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
23 rneq 5949 . . . . . . . . . . 11 (𝑓 = (𝐹𝑔) → ran 𝑓 = ran (𝐹𝑔))
24 rnco2 6274 . . . . . . . . . . 11 ran (𝐹𝑔) = (𝐹 “ ran 𝑔)
2523, 24eqtrdi 2790 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2625unieqd 4924 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ran 𝑓 = (𝐹 “ ran 𝑔))
2726, 25eleq12d 2832 . . . . . . . 8 (𝑓 = (𝐹𝑔) → ( ran 𝑓 ∈ ran 𝑓 (𝐹 “ ran 𝑔) ∈ (𝐹 “ ran 𝑔)))
2822, 27imbi12d 344 . . . . . . 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 3618 . . . . . 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 4152 . . . . . . . . 9 ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦)))
3213ffvelcdmda 7103 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ∈ 𝒫 𝐴)
3332elpwid 4613 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔𝑦) ⊆ 𝐴)
342isf34lem1 10409 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔𝑦) ⊆ 𝐴) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
3510, 33, 34syl2an2r 685 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔𝑦)) = (𝐴 ∖ (𝑔𝑦)))
36 peano2 7912 . . . . . . . . . . . . 13 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
37 ffvelcdm 7100 . . . . . . . . . . . . 13 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3813, 36, 37syl2an 596 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ∈ 𝒫 𝐴)
3938elpwid 4613 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝑔‘suc 𝑦) ⊆ 𝐴)
402isf34lem1 10409 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (𝑔‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4110, 39, 40syl2an2r 685 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (𝐹‘(𝑔‘suc 𝑦)) = (𝐴 ∖ (𝑔‘suc 𝑦)))
4235, 41sseq12d 4028 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦)) ↔ (𝐴 ∖ (𝑔𝑦)) ⊆ (𝐴 ∖ (𝑔‘suc 𝑦))))
4331, 42imbitrrid 246 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
44 fvco3 7007 . . . . . . . . . 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 7007 . . . . . . . . . 10 ((𝑔:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4713, 36, 46syl2an 596 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝐹𝑔)‘suc 𝑦) = (𝐹‘(𝑔‘suc 𝑦)))
4845, 47sseq12d 4028 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → (((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦) ↔ (𝐹‘(𝑔𝑦)) ⊆ (𝐹‘(𝑔‘suc 𝑦))))
4943, 48sylibrd 259 . . . . . . 7 ((𝑔 ∈ (𝒫 𝐴m ω) ∧ 𝑦 ∈ ω) → ((𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5049ralimdva 3164 . . . . . 6 (𝑔 ∈ (𝒫 𝐴m ω) → (∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ∀𝑦 ∈ ω ((𝐹𝑔)‘𝑦) ⊆ ((𝐹𝑔)‘suc 𝑦)))
5112ffnd 6737 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → 𝐹 Fn 𝒫 𝐴)
52 imassrn 6090 . . . . . . . . 9 (𝐹 “ ran 𝑔) ⊆ ran 𝐹
5312frnd 6744 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝐹 ⊆ 𝒫 𝐴)
5452, 53sstrid 4006 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴)
55 fnfvima 7252 . . . . . . . . 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 1120 . . . . . . . 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 4216 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
5913frnd 6744 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝑔 ⊆ 𝒫 𝐴)
6012fdmd 6746 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴m ω) → dom 𝐹 = 𝒫 𝐴)
6159, 60sseqtrrd 4036 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝑔 ⊆ dom 𝐹)
62 dfss2 3980 . . . . . . . . . . . . . 14 (ran 𝑔 ⊆ dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6361, 62sylib 218 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴m ω) → (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6458, 63eqtrid 2786 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴m ω) → (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
6513fdmd 6746 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴m ω) → dom 𝑔 = ω)
66 peano1 7910 . . . . . . . . . . . . . . 15 ∅ ∈ ω
67 ne0i 4346 . . . . . . . . . . . . . . 15 (∅ ∈ ω → ω ≠ ∅)
6866, 67mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴m ω) → ω ≠ ∅)
6965, 68eqnetrd 3005 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴m ω) → dom 𝑔 ≠ ∅)
70 dm0rn0 5937 . . . . . . . . . . . . . 14 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
7170necon3bii 2990 . . . . . . . . . . . . 13 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
7269, 71sylib 218 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴m ω) → ran 𝑔 ≠ ∅)
7364, 72eqnetrd 3005 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴m ω) → (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
74 imadisj 6099 . . . . . . . . . . . 12 ((𝐹 “ ran 𝑔) = ∅ ↔ (dom 𝐹 ∩ ran 𝑔) = ∅)
7574necon3bii 2990 . . . . . . . . . . 11 ((𝐹 “ ran 𝑔) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝑔) ≠ ∅)
7673, 75sylibr 234 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ ran 𝑔) ≠ ∅)
772isf34lem4 10414 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 “ ran 𝑔) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝑔) ≠ ∅)) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
7810, 54, 76, 77syl12anc 837 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 (𝐹 “ ran 𝑔)) = (𝐹 “ (𝐹 “ ran 𝑔)))
792isf34lem3 10412 . . . . . . . . . . 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 4955 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 “ (𝐹 “ ran 𝑔)) = ran 𝑔)
8278, 81eqtrd 2774 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴m ω) → (𝐹 (𝐹 “ ran 𝑔)) = ran 𝑔)
8382, 80eleq12d 2832 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴m ω) → ((𝐹 (𝐹 “ ran 𝑔)) ∈ (𝐹 “ (𝐹 “ ran 𝑔)) ↔ ran 𝑔 ∈ ran 𝑔))
8457, 83sylibd 239 . . . . . 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 3142 . . 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 10405 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ∀𝑔 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑔‘suc 𝑦) ⊆ (𝑔𝑦) → ran 𝑔 ∈ ran 𝑔)))
8987, 88imbitrrid 246 . 2 (𝐴𝑉 → (∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑦 ∈ ω (𝑓𝑦) ⊆ (𝑓‘suc 𝑦) → ran 𝑓 ∈ ran 𝑓) → 𝐴 ∈ FinIII))
906, 89impbid2 226 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 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cdif 3959  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604   cuni 4911   cint 4950  cmpt 5230  dom cdm 5688  ran crn 5689  cima 5691  ccom 5692  suc csuc 6387   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  ωcom 7886  m cmap 8864  FinIIIcfin3 10318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rpss 7741  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seqom 8486  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-wdom 9602  df-card 9976  df-fin4 10324  df-fin3 10325
This theorem is referenced by:  isfin3-4  10419
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