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Theorem fin1a2lem7 10400
Description: Lemma for fin1a2 10409. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b 𝐸 = (π‘₯ ∈ Ο‰ ↦ (2o Β·o π‘₯))
fin1a2lem.aa 𝑆 = (π‘₯ ∈ On ↦ suc π‘₯)
Assertion
Ref Expression
fin1a2lem7 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII)) β†’ 𝐴 ∈ FinIII)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐸
Allowed substitution hints:   𝐴(π‘₯)   𝑆(π‘₯,𝑦)   𝐸(π‘₯)   𝑉(π‘₯,𝑦)
Proof of Theorem fin1a2lem7
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 peano1 7878 . . . . . 6 βˆ… ∈ Ο‰
2 ne0i 4334 . . . . . 6 (βˆ… ∈ Ο‰ β†’ Ο‰ β‰  βˆ…)
3 brwdomn0 9563 . . . . . 6 (Ο‰ β‰  βˆ… β†’ (Ο‰ β‰Ό* 𝐴 ↔ βˆƒπ‘“ 𝑓:𝐴–ontoβ†’Ο‰))
41, 2, 3mp2b 10 . . . . 5 (Ο‰ β‰Ό* 𝐴 ↔ βˆƒπ‘“ 𝑓:𝐴–ontoβ†’Ο‰)
5 vex 3478 . . . . . . . . . 10 𝑓 ∈ V
6 fof 6805 . . . . . . . . . 10 (𝑓:𝐴–ontoβ†’Ο‰ β†’ 𝑓:π΄βŸΆΟ‰)
7 dmfex 7897 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:π΄βŸΆΟ‰) β†’ 𝐴 ∈ V)
85, 6, 7sylancr 587 . . . . . . . . 9 (𝑓:𝐴–ontoβ†’Ο‰ β†’ 𝐴 ∈ V)
9 cnvimass 6080 . . . . . . . . . 10 (◑𝑓 β€œ ran 𝐸) βŠ† dom 𝑓
109, 6fssdm 6737 . . . . . . . . 9 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (◑𝑓 β€œ ran 𝐸) βŠ† 𝐴)
118, 10sselpwd 5326 . . . . . . . 8 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (◑𝑓 β€œ ran 𝐸) ∈ 𝒫 𝐴)
12 fin1a2lem.b . . . . . . . . . . . . . 14 𝐸 = (π‘₯ ∈ Ο‰ ↦ (2o Β·o π‘₯))
1312fin1a2lem4 10397 . . . . . . . . . . . . 13 𝐸:ω–1-1β†’Ο‰
14 f1cnv 6857 . . . . . . . . . . . . 13 (𝐸:ω–1-1β†’Ο‰ β†’ ◑𝐸:ran 𝐸–1-1-ontoβ†’Ο‰)
15 f1ofo 6840 . . . . . . . . . . . . 13 (◑𝐸:ran 𝐸–1-1-ontoβ†’Ο‰ β†’ ◑𝐸:ran 𝐸–ontoβ†’Ο‰)
1613, 14, 15mp2b 10 . . . . . . . . . . . 12 ◑𝐸:ran 𝐸–ontoβ†’Ο‰
17 fofun 6806 . . . . . . . . . . . 12 (◑𝐸:ran 𝐸–ontoβ†’Ο‰ β†’ Fun ◑𝐸)
1816, 17ax-mp 5 . . . . . . . . . . 11 Fun ◑𝐸
195resex 6029 . . . . . . . . . . 11 (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)) ∈ V
20 cofunexg 7934 . . . . . . . . . . 11 ((Fun ◑𝐸 ∧ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)) ∈ V) β†’ (◑𝐸 ∘ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸))) ∈ V)
2118, 19, 20mp2an 690 . . . . . . . . . 10 (◑𝐸 ∘ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸))) ∈ V
22 fofun 6806 . . . . . . . . . . . . 13 (𝑓:𝐴–ontoβ†’Ο‰ β†’ Fun 𝑓)
23 fores 6815 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (◑𝑓 β€œ ran 𝐸) βŠ† dom 𝑓) β†’ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’(𝑓 β€œ (◑𝑓 β€œ ran 𝐸)))
2422, 9, 23sylancl 586 . . . . . . . . . . . 12 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’(𝑓 β€œ (◑𝑓 β€œ ran 𝐸)))
25 f1f 6787 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1β†’Ο‰ β†’ 𝐸:Ο‰βŸΆΟ‰)
26 frn 6724 . . . . . . . . . . . . . . 15 (𝐸:Ο‰βŸΆΟ‰ β†’ ran 𝐸 βŠ† Ο‰)
2713, 25, 26mp2b 10 . . . . . . . . . . . . . 14 ran 𝐸 βŠ† Ο‰
28 foimacnv 6850 . . . . . . . . . . . . . 14 ((𝑓:𝐴–ontoβ†’Ο‰ ∧ ran 𝐸 βŠ† Ο‰) β†’ (𝑓 β€œ (◑𝑓 β€œ ran 𝐸)) = ran 𝐸)
2927, 28mpan2 689 . . . . . . . . . . . . 13 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β€œ (◑𝑓 β€œ ran 𝐸)) = ran 𝐸)
30 foeq3 6803 . . . . . . . . . . . . 13 ((𝑓 β€œ (◑𝑓 β€œ ran 𝐸)) = ran 𝐸 β†’ ((𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’(𝑓 β€œ (◑𝑓 β€œ ran 𝐸)) ↔ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’ran 𝐸))
3129, 30syl 17 . . . . . . . . . . . 12 (𝑓:𝐴–ontoβ†’Ο‰ β†’ ((𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’(𝑓 β€œ (◑𝑓 β€œ ran 𝐸)) ↔ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’ran 𝐸))
3224, 31mpbid 231 . . . . . . . . . . 11 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’ran 𝐸)
33 foco 6819 . . . . . . . . . . 11 ((◑𝐸:ran 𝐸–ontoβ†’Ο‰ ∧ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸)):(◑𝑓 β€œ ran 𝐸)–ontoβ†’ran 𝐸) β†’ (◑𝐸 ∘ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸))):(◑𝑓 β€œ ran 𝐸)–ontoβ†’Ο‰)
3416, 32, 33sylancr 587 . . . . . . . . . 10 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (◑𝐸 ∘ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸))):(◑𝑓 β€œ ran 𝐸)–ontoβ†’Ο‰)
35 fowdom 9565 . . . . . . . . . 10 (((◑𝐸 ∘ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸))) ∈ V ∧ (◑𝐸 ∘ (𝑓 β†Ύ (◑𝑓 β€œ ran 𝐸))):(◑𝑓 β€œ ran 𝐸)–ontoβ†’Ο‰) β†’ Ο‰ β‰Ό* (◑𝑓 β€œ ran 𝐸))
3621, 34, 35sylancr 587 . . . . . . . . 9 (𝑓:𝐴–ontoβ†’Ο‰ β†’ Ο‰ β‰Ό* (◑𝑓 β€œ ran 𝐸))
375cnvex 7915 . . . . . . . . . . . 12 ◑𝑓 ∈ V
3837imaex 7906 . . . . . . . . . . 11 (◑𝑓 β€œ ran 𝐸) ∈ V
39 isfin3-2 10361 . . . . . . . . . . 11 ((◑𝑓 β€œ ran 𝐸) ∈ V β†’ ((◑𝑓 β€œ ran 𝐸) ∈ FinIII ↔ Β¬ Ο‰ β‰Ό* (◑𝑓 β€œ ran 𝐸)))
4038, 39ax-mp 5 . . . . . . . . . 10 ((◑𝑓 β€œ ran 𝐸) ∈ FinIII ↔ Β¬ Ο‰ β‰Ό* (◑𝑓 β€œ ran 𝐸))
4140con2bii 357 . . . . . . . . 9 (Ο‰ β‰Ό* (◑𝑓 β€œ ran 𝐸) ↔ Β¬ (◑𝑓 β€œ ran 𝐸) ∈ FinIII)
4236, 41sylib 217 . . . . . . . 8 (𝑓:𝐴–ontoβ†’Ο‰ β†’ Β¬ (◑𝑓 β€œ ran 𝐸) ∈ FinIII)
43 fin1a2lem.aa . . . . . . . . . . . . . . 15 𝑆 = (π‘₯ ∈ On ↦ suc π‘₯)
4412, 43fin1a2lem6 10399 . . . . . . . . . . . . . 14 (𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(Ο‰ βˆ– ran 𝐸)
45 f1ocnv 6845 . . . . . . . . . . . . . 14 ((𝑆 β†Ύ ran 𝐸):ran 𝐸–1-1-ontoβ†’(Ο‰ βˆ– ran 𝐸) β†’ β—‘(𝑆 β†Ύ ran 𝐸):(Ο‰ βˆ– ran 𝐸)–1-1-ontoβ†’ran 𝐸)
46 f1ofo 6840 . . . . . . . . . . . . . 14 (β—‘(𝑆 β†Ύ ran 𝐸):(Ο‰ βˆ– ran 𝐸)–1-1-ontoβ†’ran 𝐸 β†’ β—‘(𝑆 β†Ύ ran 𝐸):(Ο‰ βˆ– ran 𝐸)–ontoβ†’ran 𝐸)
4744, 45, 46mp2b 10 . . . . . . . . . . . . 13 β—‘(𝑆 β†Ύ ran 𝐸):(Ο‰ βˆ– ran 𝐸)–ontoβ†’ran 𝐸
48 foco 6819 . . . . . . . . . . . . 13 ((◑𝐸:ran 𝐸–ontoβ†’Ο‰ ∧ β—‘(𝑆 β†Ύ ran 𝐸):(Ο‰ βˆ– ran 𝐸)–ontoβ†’ran 𝐸) β†’ (◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)):(Ο‰ βˆ– ran 𝐸)–ontoβ†’Ο‰)
4916, 47, 48mp2an 690 . . . . . . . . . . . 12 (◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)):(Ο‰ βˆ– ran 𝐸)–ontoβ†’Ο‰
50 fofun 6806 . . . . . . . . . . . 12 ((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)):(Ο‰ βˆ– ran 𝐸)–ontoβ†’Ο‰ β†’ Fun (◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)))
5149, 50ax-mp 5 . . . . . . . . . . 11 Fun (◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸))
525resex 6029 . . . . . . . . . . 11 (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))) ∈ V
53 cofunexg 7934 . . . . . . . . . . 11 ((Fun (◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∧ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))) ∈ V) β†’ ((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∘ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))) ∈ V)
5451, 52, 53mp2an 690 . . . . . . . . . 10 ((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∘ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))) ∈ V
55 difss 4131 . . . . . . . . . . . . . 14 (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) βŠ† 𝐴
566fdmd 6728 . . . . . . . . . . . . . 14 (𝑓:𝐴–ontoβ†’Ο‰ β†’ dom 𝑓 = 𝐴)
5755, 56sseqtrrid 4035 . . . . . . . . . . . . 13 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) βŠ† dom 𝑓)
58 fores 6815 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) βŠ† dom 𝑓) β†’ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))))
5922, 57, 58syl2anc 584 . . . . . . . . . . . 12 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))))
60 funcnvcnv 6615 . . . . . . . . . . . . . . . 16 (Fun 𝑓 β†’ Fun ◑◑𝑓)
61 imadif 6632 . . . . . . . . . . . . . . . 16 (Fun ◑◑𝑓 β†’ (◑𝑓 β€œ (Ο‰ βˆ– ran 𝐸)) = ((◑𝑓 β€œ Ο‰) βˆ– (◑𝑓 β€œ ran 𝐸)))
6222, 60, 613syl 18 . . . . . . . . . . . . . . 15 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (◑𝑓 β€œ (Ο‰ βˆ– ran 𝐸)) = ((◑𝑓 β€œ Ο‰) βˆ– (◑𝑓 β€œ ran 𝐸)))
6362imaeq2d 6059 . . . . . . . . . . . . . 14 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β€œ (◑𝑓 β€œ (Ο‰ βˆ– ran 𝐸))) = (𝑓 β€œ ((◑𝑓 β€œ Ο‰) βˆ– (◑𝑓 β€œ ran 𝐸))))
64 difss 4131 . . . . . . . . . . . . . . 15 (Ο‰ βˆ– ran 𝐸) βŠ† Ο‰
65 foimacnv 6850 . . . . . . . . . . . . . . 15 ((𝑓:𝐴–ontoβ†’Ο‰ ∧ (Ο‰ βˆ– ran 𝐸) βŠ† Ο‰) β†’ (𝑓 β€œ (◑𝑓 β€œ (Ο‰ βˆ– ran 𝐸))) = (Ο‰ βˆ– ran 𝐸))
6664, 65mpan2 689 . . . . . . . . . . . . . 14 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β€œ (◑𝑓 β€œ (Ο‰ βˆ– ran 𝐸))) = (Ο‰ βˆ– ran 𝐸))
67 fimacnv 6739 . . . . . . . . . . . . . . . . 17 (𝑓:π΄βŸΆΟ‰ β†’ (◑𝑓 β€œ Ο‰) = 𝐴)
686, 67syl 17 . . . . . . . . . . . . . . . 16 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (◑𝑓 β€œ Ο‰) = 𝐴)
6968difeq1d 4121 . . . . . . . . . . . . . . 15 (𝑓:𝐴–ontoβ†’Ο‰ β†’ ((◑𝑓 β€œ Ο‰) βˆ– (◑𝑓 β€œ ran 𝐸)) = (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))
7069imaeq2d 6059 . . . . . . . . . . . . . 14 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β€œ ((◑𝑓 β€œ Ο‰) βˆ– (◑𝑓 β€œ ran 𝐸))) = (𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))))
7163, 66, 703eqtr3rd 2781 . . . . . . . . . . . . 13 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))) = (Ο‰ βˆ– ran 𝐸))
72 foeq3 6803 . . . . . . . . . . . . 13 ((𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))) = (Ο‰ βˆ– ran 𝐸) β†’ ((𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))) ↔ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(Ο‰ βˆ– ran 𝐸)))
7371, 72syl 17 . . . . . . . . . . . 12 (𝑓:𝐴–ontoβ†’Ο‰ β†’ ((𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(𝑓 β€œ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))) ↔ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(Ο‰ βˆ– ran 𝐸)))
7459, 73mpbid 231 . . . . . . . . . . 11 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(Ο‰ βˆ– ran 𝐸))
75 foco 6819 . . . . . . . . . . 11 (((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)):(Ο‰ βˆ– ran 𝐸)–ontoβ†’Ο‰ ∧ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’(Ο‰ βˆ– ran 𝐸)) β†’ ((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∘ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’Ο‰)
7649, 74, 75sylancr 587 . . . . . . . . . 10 (𝑓:𝐴–ontoβ†’Ο‰ β†’ ((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∘ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’Ο‰)
77 fowdom 9565 . . . . . . . . . 10 ((((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∘ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))) ∈ V ∧ ((◑𝐸 ∘ β—‘(𝑆 β†Ύ ran 𝐸)) ∘ (𝑓 β†Ύ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))):(𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))–ontoβ†’Ο‰) β†’ Ο‰ β‰Ό* (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))
7854, 76, 77sylancr 587 . . . . . . . . 9 (𝑓:𝐴–ontoβ†’Ο‰ β†’ Ο‰ β‰Ό* (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))
79 difexg 5327 . . . . . . . . . . 11 (𝐴 ∈ V β†’ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ V)
80 isfin3-2 10361 . . . . . . . . . . 11 ((𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ V β†’ ((𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII ↔ Β¬ Ο‰ β‰Ό* (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))))
818, 79, 803syl 18 . . . . . . . . . 10 (𝑓:𝐴–ontoβ†’Ο‰ β†’ ((𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII ↔ Β¬ Ο‰ β‰Ό* (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸))))
8281con2bid 354 . . . . . . . . 9 (𝑓:𝐴–ontoβ†’Ο‰ β†’ (Ο‰ β‰Ό* (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ↔ Β¬ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII))
8378, 82mpbid 231 . . . . . . . 8 (𝑓:𝐴–ontoβ†’Ο‰ β†’ Β¬ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII)
84 eleq1 2821 . . . . . . . . . . . 12 (𝑦 = (◑𝑓 β€œ ran 𝐸) β†’ (𝑦 ∈ FinIII ↔ (◑𝑓 β€œ ran 𝐸) ∈ FinIII))
85 difeq2 4116 . . . . . . . . . . . . 13 (𝑦 = (◑𝑓 β€œ ran 𝐸) β†’ (𝐴 βˆ– 𝑦) = (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)))
8685eleq1d 2818 . . . . . . . . . . . 12 (𝑦 = (◑𝑓 β€œ ran 𝐸) β†’ ((𝐴 βˆ– 𝑦) ∈ FinIII ↔ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII))
8784, 86orbi12d 917 . . . . . . . . . . 11 (𝑦 = (◑𝑓 β€œ ran 𝐸) β†’ ((𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII) ↔ ((◑𝑓 β€œ ran 𝐸) ∈ FinIII ∨ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII)))
8887notbid 317 . . . . . . . . . 10 (𝑦 = (◑𝑓 β€œ ran 𝐸) β†’ (Β¬ (𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII) ↔ Β¬ ((◑𝑓 β€œ ran 𝐸) ∈ FinIII ∨ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII)))
89 ioran 982 . . . . . . . . . 10 (Β¬ ((◑𝑓 β€œ ran 𝐸) ∈ FinIII ∨ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII) ↔ (Β¬ (◑𝑓 β€œ ran 𝐸) ∈ FinIII ∧ Β¬ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII))
9088, 89bitrdi 286 . . . . . . . . 9 (𝑦 = (◑𝑓 β€œ ran 𝐸) β†’ (Β¬ (𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII) ↔ (Β¬ (◑𝑓 β€œ ran 𝐸) ∈ FinIII ∧ Β¬ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII)))
9190rspcev 3612 . . . . . . . 8 (((◑𝑓 β€œ ran 𝐸) ∈ 𝒫 𝐴 ∧ (Β¬ (◑𝑓 β€œ ran 𝐸) ∈ FinIII ∧ Β¬ (𝐴 βˆ– (◑𝑓 β€œ ran 𝐸)) ∈ FinIII)) β†’ βˆƒπ‘¦ ∈ 𝒫 𝐴 Β¬ (𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII))
9211, 42, 83, 91syl12anc 835 . . . . . . 7 (𝑓:𝐴–ontoβ†’Ο‰ β†’ βˆƒπ‘¦ ∈ 𝒫 𝐴 Β¬ (𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII))
93 rexnal 3100 . . . . . . 7 (βˆƒπ‘¦ ∈ 𝒫 𝐴 Β¬ (𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII) ↔ Β¬ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII))
9492, 93sylib 217 . . . . . 6 (𝑓:𝐴–ontoβ†’Ο‰ β†’ Β¬ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII))
9594exlimiv 1933 . . . . 5 (βˆƒπ‘“ 𝑓:𝐴–ontoβ†’Ο‰ β†’ Β¬ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII))
964, 95sylbi 216 . . . 4 (Ο‰ β‰Ό* 𝐴 β†’ Β¬ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII))
9796con2i 139 . . 3 (βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII) β†’ Β¬ Ο‰ β‰Ό* 𝐴)
98 isfin3-2 10361 . . 3 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ FinIII ↔ Β¬ Ο‰ β‰Ό* 𝐴))
9997, 98imbitrrid 245 . 2 (𝐴 ∈ 𝑉 β†’ (βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII) β†’ 𝐴 ∈ FinIII))
10099imp 407 1 ((𝐴 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴 βˆ– 𝑦) ∈ FinIII)) β†’ 𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602   class class class wbr 5148   ↦ cmpt 5231  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679   ∘ ccom 5680  Oncon0 6364  suc csuc 6366  Fun wfun 6537  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  (class class class)co 7408  Ο‰com 7854  2oc2o 8459   Β·o comu 8463   β‰Ό* cwdom 9558  FinIIIcfin3 10275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seqom 8447  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-wdom 9559  df-card 9933  df-fin4 10281  df-fin3 10282
This theorem is referenced by:  fin1a2lem8  10401
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