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Theorem aomclem6 41389
Description: Lemma for dfac11 41392. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem6.b 𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}
aomclem6.c 𝐢 = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))
aomclem6.d 𝐷 = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))
aomclem6.e 𝐸 = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}
aomclem6.f 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}
aomclem6.g 𝐺 = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))
aomclem6.h 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
aomclem6.a (πœ‘ β†’ 𝐴 ∈ On)
aomclem6.y (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))
Assertion
Ref Expression
aomclem6 (πœ‘ β†’ (π»β€˜π΄) We (𝑅1β€˜π΄))
Distinct variable groups:   𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑   πœ‘,π‘Ž,𝑏,𝑐,𝑑,𝑧   𝐢,π‘Ž,𝑏,𝑐,𝑑   𝐷,π‘Ž,𝑏,𝑐,𝑑   𝐴,π‘Ž,𝑏,𝑐,𝑑,𝑧   𝐻,π‘Ž,𝑏,𝑐,𝑑,𝑧   𝐺,𝑑
Allowed substitution hints:   πœ‘(𝑦)   𝐴(𝑦)   𝐡(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐢(𝑦,𝑧)   𝐷(𝑦,𝑧)   𝐸(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐹(𝑦,𝑧,π‘Ž,𝑏,𝑐,𝑑)   𝐺(𝑦,𝑧,π‘Ž,𝑏,𝑐)   𝐻(𝑦)

Proof of Theorem aomclem6
StepHypRef Expression
1 ssid 3967 . 2 𝐴 βŠ† 𝐴
2 aomclem6.a . . . 4 (πœ‘ β†’ 𝐴 ∈ On)
32adantr 482 . . 3 ((πœ‘ ∧ 𝐴 βŠ† 𝐴) β†’ 𝐴 ∈ On)
4 sseq1 3970 . . . . . 6 (𝑐 = 𝑑 β†’ (𝑐 βŠ† 𝐴 ↔ 𝑑 βŠ† 𝐴))
54anbi2d 630 . . . . 5 (𝑐 = 𝑑 β†’ ((πœ‘ ∧ 𝑐 βŠ† 𝐴) ↔ (πœ‘ ∧ 𝑑 βŠ† 𝐴)))
6 fveq2 6843 . . . . . 6 (𝑐 = 𝑑 β†’ (π»β€˜π‘) = (π»β€˜π‘‘))
7 fveq2 6843 . . . . . 6 (𝑐 = 𝑑 β†’ (𝑅1β€˜π‘) = (𝑅1β€˜π‘‘))
86, 7weeq12d 41370 . . . . 5 (𝑐 = 𝑑 β†’ ((π»β€˜π‘) We (𝑅1β€˜π‘) ↔ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)))
95, 8imbi12d 345 . . . 4 (𝑐 = 𝑑 β†’ (((πœ‘ ∧ 𝑐 βŠ† 𝐴) β†’ (π»β€˜π‘) We (𝑅1β€˜π‘)) ↔ ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘))))
10 sseq1 3970 . . . . . 6 (𝑐 = 𝐴 β†’ (𝑐 βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
1110anbi2d 630 . . . . 5 (𝑐 = 𝐴 β†’ ((πœ‘ ∧ 𝑐 βŠ† 𝐴) ↔ (πœ‘ ∧ 𝐴 βŠ† 𝐴)))
12 fveq2 6843 . . . . . 6 (𝑐 = 𝐴 β†’ (π»β€˜π‘) = (π»β€˜π΄))
13 fveq2 6843 . . . . . 6 (𝑐 = 𝐴 β†’ (𝑅1β€˜π‘) = (𝑅1β€˜π΄))
1412, 13weeq12d 41370 . . . . 5 (𝑐 = 𝐴 β†’ ((π»β€˜π‘) We (𝑅1β€˜π‘) ↔ (π»β€˜π΄) We (𝑅1β€˜π΄)))
1511, 14imbi12d 345 . . . 4 (𝑐 = 𝐴 β†’ (((πœ‘ ∧ 𝑐 βŠ† 𝐴) β†’ (π»β€˜π‘) We (𝑅1β€˜π‘)) ↔ ((πœ‘ ∧ 𝐴 βŠ† 𝐴) β†’ (π»β€˜π΄) We (𝑅1β€˜π΄))))
16 aomclem6.b . . . . . . . . . . . . . 14 𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}
17 aomclem6.c . . . . . . . . . . . . . 14 𝐢 = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))
18 aomclem6.d . . . . . . . . . . . . . 14 𝐷 = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))
19 aomclem6.e . . . . . . . . . . . . . 14 𝐸 = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}
20 aomclem6.f . . . . . . . . . . . . . 14 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}
21 aomclem6.g . . . . . . . . . . . . . 14 𝐺 = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))
22 dmeq 5860 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝐻 β†Ύ 𝑐) β†’ dom 𝑧 = dom (𝐻 β†Ύ 𝑐))
2322adantl 483 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ dom 𝑧 = dom (𝐻 β†Ύ 𝑐))
24 simpl1 1192 . . . . . . . . . . . . . . . . 17 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ 𝑐 ∈ On)
25 onss 7720 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ On β†’ 𝑐 βŠ† On)
26 aomclem6.h . . . . . . . . . . . . . . . . . . 19 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))
2726tfr1 8344 . . . . . . . . . . . . . . . . . 18 𝐻 Fn On
28 fnssres 6625 . . . . . . . . . . . . . . . . . 18 ((𝐻 Fn On ∧ 𝑐 βŠ† On) β†’ (𝐻 β†Ύ 𝑐) Fn 𝑐)
2927, 28mpan 689 . . . . . . . . . . . . . . . . 17 (𝑐 βŠ† On β†’ (𝐻 β†Ύ 𝑐) Fn 𝑐)
30 fndm 6606 . . . . . . . . . . . . . . . . 17 ((𝐻 β†Ύ 𝑐) Fn 𝑐 β†’ dom (𝐻 β†Ύ 𝑐) = 𝑐)
3124, 25, 29, 304syl 19 . . . . . . . . . . . . . . . 16 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ dom (𝐻 β†Ύ 𝑐) = 𝑐)
3223, 31eqtrd 2777 . . . . . . . . . . . . . . 15 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ dom 𝑧 = 𝑐)
3332, 24eqeltrd 2838 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ dom 𝑧 ∈ On)
3432eleq2d 2824 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ (π‘Ž ∈ dom 𝑧 ↔ π‘Ž ∈ 𝑐))
3534biimpa 478 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ π‘Ž ∈ 𝑐)
36 simpll2 1214 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)))
37 simpl3l 1229 . . . . . . . . . . . . . . . . . 18 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ πœ‘)
3837adantr 482 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ πœ‘)
39 onelss 6360 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑧 ∈ On β†’ (π‘Ž ∈ dom 𝑧 β†’ π‘Ž βŠ† dom 𝑧))
4033, 39syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ (π‘Ž ∈ dom 𝑧 β†’ π‘Ž βŠ† dom 𝑧))
4140imp 408 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ π‘Ž βŠ† dom 𝑧)
42 simpl3r 1230 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ 𝑐 βŠ† 𝐴)
4332, 42eqsstrd 3983 . . . . . . . . . . . . . . . . . . 19 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ dom 𝑧 βŠ† 𝐴)
4443adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ dom 𝑧 βŠ† 𝐴)
4541, 44sstrd 3955 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ π‘Ž βŠ† 𝐴)
46 sseq1 3970 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = π‘Ž β†’ (𝑑 βŠ† 𝐴 ↔ π‘Ž βŠ† 𝐴))
4746anbi2d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = π‘Ž β†’ ((πœ‘ ∧ 𝑑 βŠ† 𝐴) ↔ (πœ‘ ∧ π‘Ž βŠ† 𝐴)))
48 fveq2 6843 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = π‘Ž β†’ (π»β€˜π‘‘) = (π»β€˜π‘Ž))
49 fveq2 6843 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 = π‘Ž β†’ (𝑅1β€˜π‘‘) = (𝑅1β€˜π‘Ž))
5048, 49weeq12d 41370 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = π‘Ž β†’ ((π»β€˜π‘‘) We (𝑅1β€˜π‘‘) ↔ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž)))
5147, 50imbi12d 345 . . . . . . . . . . . . . . . . . . 19 (𝑑 = π‘Ž β†’ (((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ↔ ((πœ‘ ∧ π‘Ž βŠ† 𝐴) β†’ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž))))
5251rspcva 3580 . . . . . . . . . . . . . . . . . 18 ((π‘Ž ∈ 𝑐 ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘))) β†’ ((πœ‘ ∧ π‘Ž βŠ† 𝐴) β†’ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž)))
5352imp 408 . . . . . . . . . . . . . . . . 17 (((π‘Ž ∈ 𝑐 ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘))) ∧ (πœ‘ ∧ π‘Ž βŠ† 𝐴)) β†’ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
5435, 36, 38, 45, 53syl22anc 838 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
55 fveq1 6842 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐻 β†Ύ 𝑐) β†’ (π‘§β€˜π‘Ž) = ((𝐻 β†Ύ 𝑐)β€˜π‘Ž))
5655ad2antlr 726 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ (π‘§β€˜π‘Ž) = ((𝐻 β†Ύ 𝑐)β€˜π‘Ž))
57 fvres 6862 . . . . . . . . . . . . . . . . . . 19 (π‘Ž ∈ 𝑐 β†’ ((𝐻 β†Ύ 𝑐)β€˜π‘Ž) = (π»β€˜π‘Ž))
5835, 57syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ ((𝐻 β†Ύ 𝑐)β€˜π‘Ž) = (π»β€˜π‘Ž))
5956, 58eqtrd 2777 . . . . . . . . . . . . . . . . 17 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ (π‘§β€˜π‘Ž) = (π»β€˜π‘Ž))
60 weeq1 5622 . . . . . . . . . . . . . . . . 17 ((π‘§β€˜π‘Ž) = (π»β€˜π‘Ž) β†’ ((π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž) ↔ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž)))
6159, 60syl 17 . . . . . . . . . . . . . . . 16 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ ((π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž) ↔ (π»β€˜π‘Ž) We (𝑅1β€˜π‘Ž)))
6254, 61mpbird 257 . . . . . . . . . . . . . . 15 ((((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) ∧ π‘Ž ∈ dom 𝑧) β†’ (π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
6362ralrimiva 3144 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
6437, 2syl 17 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ 𝐴 ∈ On)
65 aomclem6.y . . . . . . . . . . . . . . 15 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))
6637, 65syl 17 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))
6716, 17, 18, 19, 20, 21, 33, 63, 64, 43, 66aomclem5 41388 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ 𝐺 We (𝑅1β€˜dom 𝑧))
6832fveq2d 6847 . . . . . . . . . . . . . 14 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ (𝑅1β€˜dom 𝑧) = (𝑅1β€˜π‘))
69 weeq2 5623 . . . . . . . . . . . . . 14 ((𝑅1β€˜dom 𝑧) = (𝑅1β€˜π‘) β†’ (𝐺 We (𝑅1β€˜dom 𝑧) ↔ 𝐺 We (𝑅1β€˜π‘)))
7068, 69syl 17 . . . . . . . . . . . . 13 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ (𝐺 We (𝑅1β€˜dom 𝑧) ↔ 𝐺 We (𝑅1β€˜π‘)))
7167, 70mpbid 231 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) ∧ 𝑧 = (𝐻 β†Ύ 𝑐)) β†’ 𝐺 We (𝑅1β€˜π‘))
7271ex 414 . . . . . . . . . . 11 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ (𝑧 = (𝐻 β†Ύ 𝑐) β†’ 𝐺 We (𝑅1β€˜π‘)))
7372alrimiv 1931 . . . . . . . . . 10 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ βˆ€π‘§(𝑧 = (𝐻 β†Ύ 𝑐) β†’ 𝐺 We (𝑅1β€˜π‘)))
74 nfv 1918 . . . . . . . . . . 11 Ⅎ𝑑(𝑧 = (𝐻 β†Ύ 𝑐) β†’ 𝐺 We (𝑅1β€˜π‘))
75 nfv 1918 . . . . . . . . . . . 12 Ⅎ𝑧 𝑑 = (𝐻 β†Ύ 𝑐)
76 nfsbc1v 3760 . . . . . . . . . . . 12 Ⅎ𝑧[𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)
7775, 76nfim 1900 . . . . . . . . . . 11 Ⅎ𝑧(𝑑 = (𝐻 β†Ύ 𝑐) β†’ [𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘))
78 eqeq1 2741 . . . . . . . . . . . 12 (𝑧 = 𝑑 β†’ (𝑧 = (𝐻 β†Ύ 𝑐) ↔ 𝑑 = (𝐻 β†Ύ 𝑐)))
79 sbceq1a 3751 . . . . . . . . . . . 12 (𝑧 = 𝑑 β†’ (𝐺 We (𝑅1β€˜π‘) ↔ [𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)))
8078, 79imbi12d 345 . . . . . . . . . . 11 (𝑧 = 𝑑 β†’ ((𝑧 = (𝐻 β†Ύ 𝑐) β†’ 𝐺 We (𝑅1β€˜π‘)) ↔ (𝑑 = (𝐻 β†Ύ 𝑐) β†’ [𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘))))
8174, 77, 80cbvalv1 2338 . . . . . . . . . 10 (βˆ€π‘§(𝑧 = (𝐻 β†Ύ 𝑐) β†’ 𝐺 We (𝑅1β€˜π‘)) ↔ βˆ€π‘‘(𝑑 = (𝐻 β†Ύ 𝑐) β†’ [𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)))
8273, 81sylib 217 . . . . . . . . 9 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ βˆ€π‘‘(𝑑 = (𝐻 β†Ύ 𝑐) β†’ [𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)))
83 nfsbc1v 3760 . . . . . . . . . 10 Ⅎ𝑑[(𝐻 β†Ύ 𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)
84 fnfun 6603 . . . . . . . . . . . 12 (𝐻 Fn On β†’ Fun 𝐻)
8527, 84ax-mp 5 . . . . . . . . . . 11 Fun 𝐻
86 vex 3450 . . . . . . . . . . 11 𝑐 ∈ V
87 resfunexg 7166 . . . . . . . . . . 11 ((Fun 𝐻 ∧ 𝑐 ∈ V) β†’ (𝐻 β†Ύ 𝑐) ∈ V)
8885, 86, 87mp2an 691 . . . . . . . . . 10 (𝐻 β†Ύ 𝑐) ∈ V
89 sbceq1a 3751 . . . . . . . . . 10 (𝑑 = (𝐻 β†Ύ 𝑐) β†’ ([𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘) ↔ [(𝐻 β†Ύ 𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)))
9083, 88, 89ceqsal 3480 . . . . . . . . 9 (βˆ€π‘‘(𝑑 = (𝐻 β†Ύ 𝑐) β†’ [𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘)) ↔ [(𝐻 β†Ύ 𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘))
9182, 90sylib 217 . . . . . . . 8 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ [(𝐻 β†Ύ 𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘))
92 sbccow 3763 . . . . . . . 8 ([(𝐻 β†Ύ 𝑐) / 𝑑][𝑑 / 𝑧]𝐺 We (𝑅1β€˜π‘) ↔ [(𝐻 β†Ύ 𝑐) / 𝑧]𝐺 We (𝑅1β€˜π‘))
9391, 92sylib 217 . . . . . . 7 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ [(𝐻 β†Ύ 𝑐) / 𝑧]𝐺 We (𝑅1β€˜π‘))
94 nfcsb1v 3881 . . . . . . . . . 10 Ⅎ𝑧⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ
95 nfcv 2908 . . . . . . . . . 10 Ⅎ𝑧(𝑅1β€˜π‘)
9694, 95nfwe 5610 . . . . . . . . 9 Ⅎ𝑧⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)
97 csbeq1a 3870 . . . . . . . . . 10 (𝑧 = (𝐻 β†Ύ 𝑐) β†’ 𝐺 = ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ)
98 weeq1 5622 . . . . . . . . . 10 (𝐺 = ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ β†’ (𝐺 We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)))
9997, 98syl 17 . . . . . . . . 9 (𝑧 = (𝐻 β†Ύ 𝑐) β†’ (𝐺 We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)))
10096, 99sbciegf 3779 . . . . . . . 8 ((𝐻 β†Ύ 𝑐) ∈ V β†’ ([(𝐻 β†Ύ 𝑐) / 𝑧]𝐺 We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)))
10188, 100ax-mp 5 . . . . . . 7 ([(𝐻 β†Ύ 𝑐) / 𝑧]𝐺 We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘))
10293, 101sylib 217 . . . . . 6 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘))
103 recsval 8351 . . . . . . . . 9 (𝑐 ∈ On β†’ (recs((𝑧 ∈ V ↦ 𝐺))β€˜π‘) = ((𝑧 ∈ V ↦ 𝐺)β€˜(recs((𝑧 ∈ V ↦ 𝐺)) β†Ύ 𝑐)))
10426fveq1i 6844 . . . . . . . . 9 (π»β€˜π‘) = (recs((𝑧 ∈ V ↦ 𝐺))β€˜π‘)
105 fvex 6856 . . . . . . . . . . . . . . 15 (𝑅1β€˜dom 𝑧) ∈ V
106105, 105xpex 7688 . . . . . . . . . . . . . 14 ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)) ∈ V
107106inex2 5276 . . . . . . . . . . . . 13 (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧))) ∈ V
10821, 107eqeltri 2834 . . . . . . . . . . . 12 𝐺 ∈ V
109108csbex 5269 . . . . . . . . . . 11 ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ ∈ V
110 eqid 2737 . . . . . . . . . . . 12 (𝑧 ∈ V ↦ 𝐺) = (𝑧 ∈ V ↦ 𝐺)
111110fvmpts 6952 . . . . . . . . . . 11 (((𝐻 β†Ύ 𝑐) ∈ V ∧ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ ∈ V) β†’ ((𝑧 ∈ V ↦ 𝐺)β€˜(𝐻 β†Ύ 𝑐)) = ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ)
11288, 109, 111mp2an 691 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)β€˜(𝐻 β†Ύ 𝑐)) = ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ
11326reseq1i 5934 . . . . . . . . . . 11 (𝐻 β†Ύ 𝑐) = (recs((𝑧 ∈ V ↦ 𝐺)) β†Ύ 𝑐)
114113fveq2i 6846 . . . . . . . . . 10 ((𝑧 ∈ V ↦ 𝐺)β€˜(𝐻 β†Ύ 𝑐)) = ((𝑧 ∈ V ↦ 𝐺)β€˜(recs((𝑧 ∈ V ↦ 𝐺)) β†Ύ 𝑐))
115112, 114eqtr3i 2767 . . . . . . . . 9 ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ = ((𝑧 ∈ V ↦ 𝐺)β€˜(recs((𝑧 ∈ V ↦ 𝐺)) β†Ύ 𝑐))
116103, 104, 1153eqtr4g 2802 . . . . . . . 8 (𝑐 ∈ On β†’ (π»β€˜π‘) = ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ)
117 weeq1 5622 . . . . . . . 8 ((π»β€˜π‘) = ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ β†’ ((π»β€˜π‘) We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)))
118116, 117syl 17 . . . . . . 7 (𝑐 ∈ On β†’ ((π»β€˜π‘) We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)))
1191183ad2ant1 1134 . . . . . 6 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ ((π»β€˜π‘) We (𝑅1β€˜π‘) ↔ ⦋(𝐻 β†Ύ 𝑐) / π‘§β¦ŒπΊ We (𝑅1β€˜π‘)))
120102, 119mpbird 257 . . . . 5 ((𝑐 ∈ On ∧ βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) ∧ (πœ‘ ∧ 𝑐 βŠ† 𝐴)) β†’ (π»β€˜π‘) We (𝑅1β€˜π‘))
1211203exp 1120 . . . 4 (𝑐 ∈ On β†’ (βˆ€π‘‘ ∈ 𝑐 ((πœ‘ ∧ 𝑑 βŠ† 𝐴) β†’ (π»β€˜π‘‘) We (𝑅1β€˜π‘‘)) β†’ ((πœ‘ ∧ 𝑐 βŠ† 𝐴) β†’ (π»β€˜π‘) We (𝑅1β€˜π‘))))
1229, 15, 121tfis3 7795 . . 3 (𝐴 ∈ On β†’ ((πœ‘ ∧ 𝐴 βŠ† 𝐴) β†’ (π»β€˜π΄) We (𝑅1β€˜π΄)))
1233, 122mpcom 38 . 2 ((πœ‘ ∧ 𝐴 βŠ† 𝐴) β†’ (π»β€˜π΄) We (𝑅1β€˜π΄))
1241, 123mpan2 690 1 (πœ‘ β†’ (π»β€˜π΄) We (𝑅1β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3446  [wsbc 3740  β¦‹csb 3856   βˆ– cdif 3908   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487  π’« cpw 4561  {csn 4587  βˆͺ cuni 4866  βˆ© cint 4908   class class class wbr 5106  {copab 5168   ↦ cmpt 5189   E cep 5537   We wwe 5588   Γ— cxp 5632  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β†Ύ cres 5636   β€œ cima 5637  Oncon0 6318  suc csuc 6320  Fun wfun 6491   Fn wfn 6492  β€˜cfv 6497  recscrecs 8317  Fincfn 8884  supcsup 9377  π‘…1cr1 9699  rankcrnk 9700
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-map 8768  df-en 8885  df-fin 8888  df-sup 9379  df-r1 9701  df-rank 9702
This theorem is referenced by:  aomclem7  41390
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