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Theorem dfac12lem1 10080
Description: Lemma for dfac12 10086. (Contributed by Mario Carneiro, 29-May-2015.)
Hypotheses
Ref Expression
dfac12.1 (πœ‘ β†’ 𝐴 ∈ On)
dfac12.3 (πœ‘ β†’ 𝐹:𝒫 (harβ€˜(𝑅1β€˜π΄))–1-1β†’On)
dfac12.4 𝐺 = recs((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦))))))
dfac12.5 (πœ‘ β†’ 𝐢 ∈ On)
dfac12.h 𝐻 = (β—‘OrdIso( E , ran (πΊβ€˜βˆͺ 𝐢)) ∘ (πΊβ€˜βˆͺ 𝐢))
Assertion
Ref Expression
dfac12lem1 (πœ‘ β†’ (πΊβ€˜πΆ) = (𝑦 ∈ (𝑅1β€˜πΆ) ↦ if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜(𝐻 β€œ 𝑦)))))
Distinct variable groups:   𝑦,𝐴   π‘₯,𝑦,𝐢   π‘₯,𝐺,𝑦   πœ‘,𝑦   π‘₯,𝐹,𝑦   𝑦,𝐻
Allowed substitution hints:   πœ‘(π‘₯)   𝐴(π‘₯)   𝐻(π‘₯)

Proof of Theorem dfac12lem1
StepHypRef Expression
1 dfac12.5 . . 3 (πœ‘ β†’ 𝐢 ∈ On)
2 dfac12.4 . . . 4 𝐺 = recs((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦))))))
32tfr2 8345 . . 3 (𝐢 ∈ On β†’ (πΊβ€˜πΆ) = ((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)))))β€˜(𝐺 β†Ύ 𝐢)))
41, 3syl 17 . 2 (πœ‘ β†’ (πΊβ€˜πΆ) = ((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)))))β€˜(𝐺 β†Ύ 𝐢)))
52tfr1 8344 . . . . 5 𝐺 Fn On
6 fnfun 6603 . . . . 5 (𝐺 Fn On β†’ Fun 𝐺)
75, 6ax-mp 5 . . . 4 Fun 𝐺
8 resfunexg 7166 . . . 4 ((Fun 𝐺 ∧ 𝐢 ∈ On) β†’ (𝐺 β†Ύ 𝐢) ∈ V)
97, 1, 8sylancr 588 . . 3 (πœ‘ β†’ (𝐺 β†Ύ 𝐢) ∈ V)
10 dmeq 5860 . . . . . 6 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ dom π‘₯ = dom (𝐺 β†Ύ 𝐢))
1110fveq2d 6847 . . . . 5 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (𝑅1β€˜dom π‘₯) = (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)))
1210unieqd 4880 . . . . . . 7 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ βˆͺ dom π‘₯ = βˆͺ dom (𝐺 β†Ύ 𝐢))
1310, 12eqeq12d 2753 . . . . . 6 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (dom π‘₯ = βˆͺ dom π‘₯ ↔ dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢)))
14 rneq 5892 . . . . . . . . . . . . 13 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran π‘₯ = ran (𝐺 β†Ύ 𝐢))
15 df-ima 5647 . . . . . . . . . . . . 13 (𝐺 β€œ 𝐢) = ran (𝐺 β†Ύ 𝐢)
1614, 15eqtr4di 2795 . . . . . . . . . . . 12 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran π‘₯ = (𝐺 β€œ 𝐢))
1716unieqd 4880 . . . . . . . . . . 11 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ βˆͺ ran π‘₯ = βˆͺ (𝐺 β€œ 𝐢))
1817rneqd 5894 . . . . . . . . . 10 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran βˆͺ ran π‘₯ = ran βˆͺ (𝐺 β€œ 𝐢))
1918unieqd 4880 . . . . . . . . 9 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ βˆͺ ran βˆͺ ran π‘₯ = βˆͺ ran βˆͺ (𝐺 β€œ 𝐢))
20 suceq 6384 . . . . . . . . 9 (βˆͺ ran βˆͺ ran π‘₯ = βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) β†’ suc βˆͺ ran βˆͺ ran π‘₯ = suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢))
2119, 20syl 17 . . . . . . . 8 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ suc βˆͺ ran βˆͺ ran π‘₯ = suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢))
2221oveq1d 7373 . . . . . . 7 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) = (suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)))
23 fveq1 6842 . . . . . . . 8 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (π‘₯β€˜suc (rankβ€˜π‘¦)) = ((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦)))
2423fveq1d 6845 . . . . . . 7 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦) = (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦))
2522, 24oveq12d 7376 . . . . . 6 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)) = ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)))
26 id 22 . . . . . . . . . . . . 13 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ π‘₯ = (𝐺 β†Ύ 𝐢))
2726, 12fveq12d 6850 . . . . . . . . . . . 12 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (π‘₯β€˜βˆͺ dom π‘₯) = ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)))
2827rneqd 5894 . . . . . . . . . . 11 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran (π‘₯β€˜βˆͺ dom π‘₯) = ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)))
29 oieq2 9450 . . . . . . . . . . 11 (ran (π‘₯β€˜βˆͺ dom π‘₯) = ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)) β†’ OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) = OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))))
3028, 29syl 17 . . . . . . . . . 10 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) = OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))))
3130cnveqd 5832 . . . . . . . . 9 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) = β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))))
3231, 27coeq12d 5821 . . . . . . . 8 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) = (β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))))
3332imaeq1d 6013 . . . . . . 7 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦) = ((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))
3433fveq2d 6847 . . . . . 6 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)) = (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))
3513, 25, 34ifbieq12d 4515 . . . . 5 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦))) = if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))))
3611, 35mpteq12dv 5197 . . . 4 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)))) = (𝑦 ∈ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))))
37 eqid 2737 . . . 4 (π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦))))) = (π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)))))
38 fvex 6856 . . . . 5 (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ∈ V
3938mptex 7174 . . . 4 (𝑦 ∈ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))) ∈ V
4036, 37, 39fvmpt 6949 . . 3 ((𝐺 β†Ύ 𝐢) ∈ V β†’ ((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)))))β€˜(𝐺 β†Ύ 𝐢)) = (𝑦 ∈ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))))
419, 40syl 17 . 2 (πœ‘ β†’ ((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦)))))β€˜(𝐺 β†Ύ 𝐢)) = (𝑦 ∈ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))))
42 onss 7720 . . . . . . . 8 (𝐢 ∈ On β†’ 𝐢 βŠ† On)
431, 42syl 17 . . . . . . 7 (πœ‘ β†’ 𝐢 βŠ† On)
44 fnssres 6625 . . . . . . 7 ((𝐺 Fn On ∧ 𝐢 βŠ† On) β†’ (𝐺 β†Ύ 𝐢) Fn 𝐢)
455, 43, 44sylancr 588 . . . . . 6 (πœ‘ β†’ (𝐺 β†Ύ 𝐢) Fn 𝐢)
4645fndmd 6608 . . . . 5 (πœ‘ β†’ dom (𝐺 β†Ύ 𝐢) = 𝐢)
4746fveq2d 6847 . . . 4 (πœ‘ β†’ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) = (𝑅1β€˜πΆ))
4847mpteq1d 5201 . . 3 (πœ‘ β†’ (𝑦 ∈ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))) = (𝑦 ∈ (𝑅1β€˜πΆ) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))))
4946adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ dom (𝐺 β†Ύ 𝐢) = 𝐢)
5049unieqd 4880 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ βˆͺ dom (𝐺 β†Ύ 𝐢) = βˆͺ 𝐢)
5149, 50eqeq12d 2753 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ (dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢) ↔ 𝐢 = βˆͺ 𝐢))
5251ifbid 4510 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))) = if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))))
53 rankr1ai 9735 . . . . . . . . . . . 12 (𝑦 ∈ (𝑅1β€˜πΆ) β†’ (rankβ€˜π‘¦) ∈ 𝐢)
5453ad2antlr 726 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ (rankβ€˜π‘¦) ∈ 𝐢)
55 simpr 486 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ 𝐢 = βˆͺ 𝐢)
5654, 55eleqtrd 2840 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ (rankβ€˜π‘¦) ∈ βˆͺ 𝐢)
57 eloni 6328 . . . . . . . . . . . 12 (𝐢 ∈ On β†’ Ord 𝐢)
58 ordsucuniel 7760 . . . . . . . . . . . 12 (Ord 𝐢 β†’ ((rankβ€˜π‘¦) ∈ βˆͺ 𝐢 ↔ suc (rankβ€˜π‘¦) ∈ 𝐢))
591, 57, 583syl 18 . . . . . . . . . . 11 (πœ‘ β†’ ((rankβ€˜π‘¦) ∈ βˆͺ 𝐢 ↔ suc (rankβ€˜π‘¦) ∈ 𝐢))
6059ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ ((rankβ€˜π‘¦) ∈ βˆͺ 𝐢 ↔ suc (rankβ€˜π‘¦) ∈ 𝐢))
6156, 60mpbid 231 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ suc (rankβ€˜π‘¦) ∈ 𝐢)
6261fvresd 6863 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ ((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦)) = (πΊβ€˜suc (rankβ€˜π‘¦)))
6362fveq1d 6845 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦) = ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦))
6463oveq2d 7374 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)) = ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)))
6564ifeq1da 4518 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))) = if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))))
6650adantr 482 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ βˆͺ dom (𝐺 β†Ύ 𝐢) = βˆͺ 𝐢)
6766fveq2d 6847 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)) = ((𝐺 β†Ύ 𝐢)β€˜βˆͺ 𝐢))
681ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ 𝐢 ∈ On)
69 uniexg 7678 . . . . . . . . . . . . . . . . 17 (𝐢 ∈ On β†’ βˆͺ 𝐢 ∈ V)
70 sucidg 6399 . . . . . . . . . . . . . . . . 17 (βˆͺ 𝐢 ∈ V β†’ βˆͺ 𝐢 ∈ suc βˆͺ 𝐢)
7168, 69, 703syl 18 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ βˆͺ 𝐢 ∈ suc βˆͺ 𝐢)
721adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ 𝐢 ∈ On)
73 orduniorsuc 7766 . . . . . . . . . . . . . . . . . 18 (Ord 𝐢 β†’ (𝐢 = βˆͺ 𝐢 ∨ 𝐢 = suc βˆͺ 𝐢))
7472, 57, 733syl 18 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ (𝐢 = βˆͺ 𝐢 ∨ 𝐢 = suc βˆͺ 𝐢))
7574orcanai 1002 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ 𝐢 = suc βˆͺ 𝐢)
7671, 75eleqtrrd 2841 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ βˆͺ 𝐢 ∈ 𝐢)
7776fvresd 6863 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ 𝐢) = (πΊβ€˜βˆͺ 𝐢))
7867, 77eqtrd 2777 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)) = (πΊβ€˜βˆͺ 𝐢))
7978rneqd 5894 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)) = ran (πΊβ€˜βˆͺ 𝐢))
80 oieq2 9450 . . . . . . . . . . . 12 (ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)) = ran (πΊβ€˜βˆͺ 𝐢) β†’ OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) = OrdIso( E , ran (πΊβ€˜βˆͺ 𝐢)))
8179, 80syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) = OrdIso( E , ran (πΊβ€˜βˆͺ 𝐢)))
8281cnveqd 5832 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) = β—‘OrdIso( E , ran (πΊβ€˜βˆͺ 𝐢)))
8382, 78coeq12d 5821 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ (β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) = (β—‘OrdIso( E , ran (πΊβ€˜βˆͺ 𝐢)) ∘ (πΊβ€˜βˆͺ 𝐢)))
84 dfac12.h . . . . . . . . 9 𝐻 = (β—‘OrdIso( E , ran (πΊβ€˜βˆͺ 𝐢)) ∘ (πΊβ€˜βˆͺ 𝐢))
8583, 84eqtr4di 2795 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ (β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) = 𝐻)
8685imaeq1d 6013 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ ((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦) = (𝐻 β€œ 𝑦))
8786fveq2d 6847 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)) = (πΉβ€˜(𝐻 β€œ 𝑦)))
8887ifeq2da 4519 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))) = if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜(𝐻 β€œ 𝑦))))
8952, 65, 883eqtrd 2781 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦))) = if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜(𝐻 β€œ 𝑦))))
9089mpteq2dva 5206 . . 3 (πœ‘ β†’ (𝑦 ∈ (𝑅1β€˜πΆ) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))) = (𝑦 ∈ (𝑅1β€˜πΆ) ↦ if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜(𝐻 β€œ 𝑦)))))
9148, 90eqtrd 2777 . 2 (πœ‘ β†’ (𝑦 ∈ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ↦ if(dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢), ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o (((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) ∘ ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢))) β€œ 𝑦)))) = (𝑦 ∈ (𝑅1β€˜πΆ) ↦ if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜(𝐻 β€œ 𝑦)))))
924, 41, 913eqtrd 2781 1 (πœ‘ β†’ (πΊβ€˜πΆ) = (𝑦 ∈ (𝑅1β€˜πΆ) ↦ if(𝐢 = βˆͺ 𝐢, ((suc βˆͺ ran βˆͺ (𝐺 β€œ 𝐢) Β·o (rankβ€˜π‘¦)) +o ((πΊβ€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜(𝐻 β€œ 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3446   βŠ† wss 3911  ifcif 4487  π’« cpw 4561  βˆͺ cuni 4866   ↦ cmpt 5189   E cep 5537  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β†Ύ cres 5636   β€œ cima 5637   ∘ ccom 5638  Ord word 6317  Oncon0 6318  suc csuc 6320  Fun wfun 6491   Fn wfn 6492  β€“1-1β†’wf1 6494  β€˜cfv 6497  (class class class)co 7358  recscrecs 8317   +o coa 8410   Β·o comu 8411  OrdIsocoi 9446  harchar 9493  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-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-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-se 5590  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-riota 7314  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-oi 9447  df-r1 9701  df-rank 9702
This theorem is referenced by:  dfac12lem2  10081
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