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Theorem isf34lem6 10375
Description: Lemma for isfin3-4 10377. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a 𝐹 = (π‘₯ ∈ 𝒫 𝐴 ↦ (𝐴 βˆ– π‘₯))
Assertion
Ref Expression
isf34lem6 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ FinIII ↔ βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓)))
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐴   𝑓,𝐹,𝑦   π‘₯,𝑉,𝑦
Allowed substitution hints:   𝐹(π‘₯)   𝑉(𝑓)

Proof of Theorem isf34lem6
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elmapi 8843 . . . 4 (𝑓 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝑓:Ο‰βŸΆπ’« 𝐴)
2 compss.a . . . . . 6 𝐹 = (π‘₯ ∈ 𝒫 𝐴 ↦ (𝐴 βˆ– π‘₯))
32isf34lem7 10374 . . . . 5 ((𝐴 ∈ FinIII ∧ 𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦)) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓)
433expia 1122 . . . 4 ((𝐴 ∈ FinIII ∧ 𝑓:Ο‰βŸΆπ’« 𝐴) β†’ (βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓))
51, 4sylan2 594 . . 3 ((𝐴 ∈ FinIII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m Ο‰)) β†’ (βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓))
65ralrimiva 3147 . 2 (𝐴 ∈ FinIII β†’ βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓))
7 elmapex 8842 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (𝒫 𝐴 ∈ V ∧ Ο‰ ∈ V))
87simpld 496 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝒫 𝐴 ∈ V)
9 pwexb 7753 . . . . . . . . . 10 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
108, 9sylibr 233 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝐴 ∈ V)
112isf34lem2 10368 . . . . . . . . 9 (𝐴 ∈ V β†’ 𝐹:𝒫 π΄βŸΆπ’« 𝐴)
1210, 11syl 17 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝐹:𝒫 π΄βŸΆπ’« 𝐴)
13 elmapi 8843 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝑔:Ο‰βŸΆπ’« 𝐴)
14 fco 6742 . . . . . . . 8 ((𝐹:𝒫 π΄βŸΆπ’« 𝐴 ∧ 𝑔:Ο‰βŸΆπ’« 𝐴) β†’ (𝐹 ∘ 𝑔):Ο‰βŸΆπ’« 𝐴)
1512, 13, 14syl2anc 585 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (𝐹 ∘ 𝑔):Ο‰βŸΆπ’« 𝐴)
16 elmapg 8833 . . . . . . . 8 ((𝒫 𝐴 ∈ V ∧ Ο‰ ∈ V) β†’ ((𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑m Ο‰) ↔ (𝐹 ∘ 𝑔):Ο‰βŸΆπ’« 𝐴))
177, 16syl 17 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ((𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑m Ο‰) ↔ (𝐹 ∘ 𝑔):Ο‰βŸΆπ’« 𝐴))
1815, 17mpbird 257 . . . . . 6 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑m Ο‰))
19 fveq1 6891 . . . . . . . . . 10 (𝑓 = (𝐹 ∘ 𝑔) β†’ (π‘“β€˜π‘¦) = ((𝐹 ∘ 𝑔)β€˜π‘¦))
20 fveq1 6891 . . . . . . . . . 10 (𝑓 = (𝐹 ∘ 𝑔) β†’ (π‘“β€˜suc 𝑦) = ((𝐹 ∘ 𝑔)β€˜suc 𝑦))
2119, 20sseq12d 4016 . . . . . . . . 9 (𝑓 = (𝐹 ∘ 𝑔) β†’ ((π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) ↔ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦)))
2221ralbidv 3178 . . . . . . . 8 (𝑓 = (𝐹 ∘ 𝑔) β†’ (βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) ↔ βˆ€π‘¦ ∈ Ο‰ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦)))
23 rneq 5936 . . . . . . . . . . 11 (𝑓 = (𝐹 ∘ 𝑔) β†’ ran 𝑓 = ran (𝐹 ∘ 𝑔))
24 rnco2 6253 . . . . . . . . . . 11 ran (𝐹 ∘ 𝑔) = (𝐹 β€œ ran 𝑔)
2523, 24eqtrdi 2789 . . . . . . . . . 10 (𝑓 = (𝐹 ∘ 𝑔) β†’ ran 𝑓 = (𝐹 β€œ ran 𝑔))
2625unieqd 4923 . . . . . . . . 9 (𝑓 = (𝐹 ∘ 𝑔) β†’ βˆͺ ran 𝑓 = βˆͺ (𝐹 β€œ ran 𝑔))
2726, 25eleq12d 2828 . . . . . . . 8 (𝑓 = (𝐹 ∘ 𝑔) β†’ (βˆͺ ran 𝑓 ∈ ran 𝑓 ↔ βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔)))
2822, 27imbi12d 345 . . . . . . 7 (𝑓 = (𝐹 ∘ 𝑔) β†’ ((βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓) ↔ (βˆ€π‘¦ ∈ Ο‰ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦) β†’ βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔))))
2928rspccv 3610 . . . . . 6 (βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓) β†’ ((𝐹 ∘ 𝑔) ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (βˆ€π‘¦ ∈ Ο‰ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦) β†’ βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔))))
3018, 29syl5 34 . . . . 5 (βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓) β†’ (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (βˆ€π‘¦ ∈ Ο‰ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦) β†’ βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔))))
31 sscon 4139 . . . . . . . . 9 ((π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ (𝐴 βˆ– (π‘”β€˜π‘¦)) βŠ† (𝐴 βˆ– (π‘”β€˜suc 𝑦)))
3213ffvelcdmda 7087 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (π‘”β€˜π‘¦) ∈ 𝒫 𝐴)
3332elpwid 4612 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (π‘”β€˜π‘¦) βŠ† 𝐴)
342isf34lem1 10367 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (π‘”β€˜π‘¦) βŠ† 𝐴) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = (𝐴 βˆ– (π‘”β€˜π‘¦)))
3510, 33, 34syl2an2r 684 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = (𝐴 βˆ– (π‘”β€˜π‘¦)))
36 peano2 7881 . . . . . . . . . . . . 13 (𝑦 ∈ Ο‰ β†’ suc 𝑦 ∈ Ο‰)
37 ffvelcdm 7084 . . . . . . . . . . . . 13 ((𝑔:Ο‰βŸΆπ’« 𝐴 ∧ suc 𝑦 ∈ Ο‰) β†’ (π‘”β€˜suc 𝑦) ∈ 𝒫 𝐴)
3813, 36, 37syl2an 597 . . . . . . . . . . . 12 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (π‘”β€˜suc 𝑦) ∈ 𝒫 𝐴)
3938elpwid 4612 . . . . . . . . . . 11 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (π‘”β€˜suc 𝑦) βŠ† 𝐴)
402isf34lem1 10367 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ (π‘”β€˜suc 𝑦) βŠ† 𝐴) β†’ (πΉβ€˜(π‘”β€˜suc 𝑦)) = (𝐴 βˆ– (π‘”β€˜suc 𝑦)))
4110, 39, 40syl2an2r 684 . . . . . . . . . 10 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (πΉβ€˜(π‘”β€˜suc 𝑦)) = (𝐴 βˆ– (π‘”β€˜suc 𝑦)))
4235, 41sseq12d 4016 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ ((πΉβ€˜(π‘”β€˜π‘¦)) βŠ† (πΉβ€˜(π‘”β€˜suc 𝑦)) ↔ (𝐴 βˆ– (π‘”β€˜π‘¦)) βŠ† (𝐴 βˆ– (π‘”β€˜suc 𝑦))))
4331, 42imbitrrid 245 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ ((π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) βŠ† (πΉβ€˜(π‘”β€˜suc 𝑦))))
44 fvco3 6991 . . . . . . . . . 10 ((𝑔:Ο‰βŸΆπ’« 𝐴 ∧ 𝑦 ∈ Ο‰) β†’ ((𝐹 ∘ 𝑔)β€˜π‘¦) = (πΉβ€˜(π‘”β€˜π‘¦)))
4513, 44sylan 581 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ ((𝐹 ∘ 𝑔)β€˜π‘¦) = (πΉβ€˜(π‘”β€˜π‘¦)))
46 fvco3 6991 . . . . . . . . . 10 ((𝑔:Ο‰βŸΆπ’« 𝐴 ∧ suc 𝑦 ∈ Ο‰) β†’ ((𝐹 ∘ 𝑔)β€˜suc 𝑦) = (πΉβ€˜(π‘”β€˜suc 𝑦)))
4713, 36, 46syl2an 597 . . . . . . . . 9 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ ((𝐹 ∘ 𝑔)β€˜suc 𝑦) = (πΉβ€˜(π‘”β€˜suc 𝑦)))
4845, 47sseq12d 4016 . . . . . . . 8 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ (((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦) ↔ (πΉβ€˜(π‘”β€˜π‘¦)) βŠ† (πΉβ€˜(π‘”β€˜suc 𝑦))))
4943, 48sylibrd 259 . . . . . . 7 ((𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) ∧ 𝑦 ∈ Ο‰) β†’ ((π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦)))
5049ralimdva 3168 . . . . . 6 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (βˆ€π‘¦ ∈ Ο‰ (π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ βˆ€π‘¦ ∈ Ο‰ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦)))
5112ffnd 6719 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝐹 Fn 𝒫 𝐴)
52 imassrn 6071 . . . . . . . . 9 (𝐹 β€œ ran 𝑔) βŠ† ran 𝐹
5312frnd 6726 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ran 𝐹 βŠ† 𝒫 𝐴)
5452, 53sstrid 3994 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (𝐹 β€œ ran 𝑔) βŠ† 𝒫 𝐴)
55 fnfvima 7235 . . . . . . . . 9 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 β€œ ran 𝑔) βŠ† 𝒫 𝐴 ∧ βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔)) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) ∈ (𝐹 β€œ (𝐹 β€œ ran 𝑔)))
56553expia 1122 . . . . . . . 8 ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 β€œ ran 𝑔) βŠ† 𝒫 𝐴) β†’ (βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) ∈ (𝐹 β€œ (𝐹 β€œ ran 𝑔))))
5751, 54, 56syl2anc 585 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) ∈ (𝐹 β€œ (𝐹 β€œ ran 𝑔))))
58 incom 4202 . . . . . . . . . . . . 13 (dom 𝐹 ∩ ran 𝑔) = (ran 𝑔 ∩ dom 𝐹)
5913frnd 6726 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ran 𝑔 βŠ† 𝒫 𝐴)
6012fdmd 6729 . . . . . . . . . . . . . . 15 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ dom 𝐹 = 𝒫 𝐴)
6159, 60sseqtrrd 4024 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ran 𝑔 βŠ† dom 𝐹)
62 df-ss 3966 . . . . . . . . . . . . . 14 (ran 𝑔 βŠ† dom 𝐹 ↔ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6361, 62sylib 217 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (ran 𝑔 ∩ dom 𝐹) = ran 𝑔)
6458, 63eqtrid 2785 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (dom 𝐹 ∩ ran 𝑔) = ran 𝑔)
6513fdmd 6729 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ dom 𝑔 = Ο‰)
66 peano1 7879 . . . . . . . . . . . . . . 15 βˆ… ∈ Ο‰
67 ne0i 4335 . . . . . . . . . . . . . . 15 (βˆ… ∈ Ο‰ β†’ Ο‰ β‰  βˆ…)
6866, 67mp1i 13 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ Ο‰ β‰  βˆ…)
6965, 68eqnetrd 3009 . . . . . . . . . . . . 13 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ dom 𝑔 β‰  βˆ…)
70 dm0rn0 5925 . . . . . . . . . . . . . 14 (dom 𝑔 = βˆ… ↔ ran 𝑔 = βˆ…)
7170necon3bii 2994 . . . . . . . . . . . . 13 (dom 𝑔 β‰  βˆ… ↔ ran 𝑔 β‰  βˆ…)
7269, 71sylib 217 . . . . . . . . . . . 12 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ran 𝑔 β‰  βˆ…)
7364, 72eqnetrd 3009 . . . . . . . . . . 11 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (dom 𝐹 ∩ ran 𝑔) β‰  βˆ…)
74 imadisj 6080 . . . . . . . . . . . 12 ((𝐹 β€œ ran 𝑔) = βˆ… ↔ (dom 𝐹 ∩ ran 𝑔) = βˆ…)
7574necon3bii 2994 . . . . . . . . . . 11 ((𝐹 β€œ ran 𝑔) β‰  βˆ… ↔ (dom 𝐹 ∩ ran 𝑔) β‰  βˆ…)
7673, 75sylibr 233 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (𝐹 β€œ ran 𝑔) β‰  βˆ…)
772isf34lem4 10372 . . . . . . . . . 10 ((𝐴 ∈ V ∧ ((𝐹 β€œ ran 𝑔) βŠ† 𝒫 𝐴 ∧ (𝐹 β€œ ran 𝑔) β‰  βˆ…)) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) = ∩ (𝐹 β€œ (𝐹 β€œ ran 𝑔)))
7810, 54, 76, 77syl12anc 836 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) = ∩ (𝐹 β€œ (𝐹 β€œ ran 𝑔)))
792isf34lem3 10370 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ ran 𝑔 βŠ† 𝒫 𝐴) β†’ (𝐹 β€œ (𝐹 β€œ ran 𝑔)) = ran 𝑔)
8010, 59, 79syl2anc 585 . . . . . . . . . 10 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (𝐹 β€œ (𝐹 β€œ ran 𝑔)) = ran 𝑔)
8180inteqd 4956 . . . . . . . . 9 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ∩ (𝐹 β€œ (𝐹 β€œ ran 𝑔)) = ∩ ran 𝑔)
8278, 81eqtrd 2773 . . . . . . . 8 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) = ∩ ran 𝑔)
8382, 80eleq12d 2828 . . . . . . 7 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ((πΉβ€˜βˆͺ (𝐹 β€œ ran 𝑔)) ∈ (𝐹 β€œ (𝐹 β€œ ran 𝑔)) ↔ ∩ ran 𝑔 ∈ ran 𝑔))
8457, 83sylibd 238 . . . . . 6 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔) β†’ ∩ ran 𝑔 ∈ ran 𝑔))
8550, 84imim12d 81 . . . . 5 (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ ((βˆ€π‘¦ ∈ Ο‰ ((𝐹 ∘ 𝑔)β€˜π‘¦) βŠ† ((𝐹 ∘ 𝑔)β€˜suc 𝑦) β†’ βˆͺ (𝐹 β€œ ran 𝑔) ∈ (𝐹 β€œ ran 𝑔)) β†’ (βˆ€π‘¦ ∈ Ο‰ (π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ ∩ ran 𝑔 ∈ ran 𝑔)))
8630, 85sylcom 30 . . . 4 (βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓) β†’ (𝑔 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ (βˆ€π‘¦ ∈ Ο‰ (π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ ∩ ran 𝑔 ∈ ran 𝑔)))
8786ralrimiv 3146 . . 3 (βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓) β†’ βˆ€π‘” ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ ∩ ran 𝑔 ∈ ran 𝑔))
88 isfin3-3 10363 . . 3 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ FinIII ↔ βˆ€π‘” ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘”β€˜suc 𝑦) βŠ† (π‘”β€˜π‘¦) β†’ ∩ ran 𝑔 ∈ ran 𝑔)))
8987, 88imbitrrid 245 . 2 (𝐴 ∈ 𝑉 β†’ (βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓) β†’ 𝐴 ∈ FinIII))
906, 89impbid2 225 1 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ FinIII ↔ βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘¦ ∈ Ο‰ (π‘“β€˜π‘¦) βŠ† (π‘“β€˜suc 𝑦) β†’ βˆͺ ran 𝑓 ∈ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  βˆ© cint 4951   ↦ cmpt 5232  dom cdm 5677  ran crn 5678   β€œ cima 5680   ∘ ccom 5681  suc csuc 6367   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855   ↑m cmap 8820  FinIIIcfin3 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rpss 7713  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-seqom 8448  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-wdom 9560  df-card 9934  df-fin4 10282  df-fin3 10283
This theorem is referenced by:  isfin3-4  10377
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