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Theorem dfac12lem1 10135
Description: Lemma for dfac12 10141. (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 8395 . . 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 8394 . . . . 5 𝐺 Fn On
6 fnfun 6647 . . . . 5 (𝐺 Fn On β†’ Fun 𝐺)
75, 6ax-mp 5 . . . 4 Fun 𝐺
8 resfunexg 7214 . . . 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 5902 . . . . . 6 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ dom π‘₯ = dom (𝐺 β†Ύ 𝐢))
1110fveq2d 6893 . . . . 5 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (𝑅1β€˜dom π‘₯) = (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)))
1210unieqd 4922 . . . . . . 7 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ βˆͺ dom π‘₯ = βˆͺ dom (𝐺 β†Ύ 𝐢))
1310, 12eqeq12d 2749 . . . . . 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 5934 . . . . . . . . . . . . 13 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran π‘₯ = ran (𝐺 β†Ύ 𝐢))
15 df-ima 5689 . . . . . . . . . . . . 13 (𝐺 β€œ 𝐢) = ran (𝐺 β†Ύ 𝐢)
1614, 15eqtr4di 2791 . . . . . . . . . . . 12 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran π‘₯ = (𝐺 β€œ 𝐢))
1716unieqd 4922 . . . . . . . . . . 11 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ βˆͺ ran π‘₯ = βˆͺ (𝐺 β€œ 𝐢))
1817rneqd 5936 . . . . . . . . . 10 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ ran βˆͺ ran π‘₯ = ran βˆͺ (𝐺 β€œ 𝐢))
1918unieqd 4922 . . . . . . . . 9 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ βˆͺ ran βˆͺ ran π‘₯ = βˆͺ ran βˆͺ (𝐺 β€œ 𝐢))
20 suceq 6428 . . . . . . . . 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 7421 . . . . . . 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 6888 . . . . . . . 8 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (π‘₯β€˜suc (rankβ€˜π‘¦)) = ((𝐺 β†Ύ 𝐢)β€˜suc (rankβ€˜π‘¦)))
2423fveq1d 6891 . . . . . . 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 7424 . . . . . 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 6896 . . . . . . . . . . . 12 (π‘₯ = (𝐺 β†Ύ 𝐢) β†’ (π‘₯β€˜βˆͺ dom π‘₯) = ((𝐺 β†Ύ 𝐢)β€˜βˆͺ dom (𝐺 β†Ύ 𝐢)))
2827rneqd 5936 . . . . . . . . . . 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 9505 . . . . . . . . . . 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 5874 . . . . . . . . 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 5863 . . . . . . . 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 6057 . . . . . . 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 6893 . . . . . 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 4556 . . . . 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 5239 . . . 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 2733 . . . 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 6902 . . . . 5 (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) ∈ V
3938mptex 7222 . . . 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 6996 . . 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 7769 . . . . . . . 8 (𝐢 ∈ On β†’ 𝐢 βŠ† On)
431, 42syl 17 . . . . . . 7 (πœ‘ β†’ 𝐢 βŠ† On)
44 fnssres 6671 . . . . . . 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 6652 . . . . 5 (πœ‘ β†’ dom (𝐺 β†Ύ 𝐢) = 𝐢)
4746fveq2d 6893 . . . 4 (πœ‘ β†’ (𝑅1β€˜dom (𝐺 β†Ύ 𝐢)) = (𝑅1β€˜πΆ))
4847mpteq1d 5243 . . 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 4922 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ βˆͺ dom (𝐺 β†Ύ 𝐢) = βˆͺ 𝐢)
5149, 50eqeq12d 2749 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) β†’ (dom (𝐺 β†Ύ 𝐢) = βˆͺ dom (𝐺 β†Ύ 𝐢) ↔ 𝐢 = βˆͺ 𝐢))
5251ifbid 4551 . . . . 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 9790 . . . . . . . . . . . 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 2836 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ 𝐢 = βˆͺ 𝐢) β†’ (rankβ€˜π‘¦) ∈ βˆͺ 𝐢)
57 eloni 6372 . . . . . . . . . . . 12 (𝐢 ∈ On β†’ Ord 𝐢)
58 ordsucuniel 7809 . . . . . . . . . . . 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 6909 . . . . . . . 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 6891 . . . . . . 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 7422 . . . . . 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 4559 . . . . 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 6893 . . . . . . . . . . . . . 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 7727 . . . . . . . . . . . . . . . . 17 (𝐢 ∈ On β†’ βˆͺ 𝐢 ∈ V)
70 sucidg 6443 . . . . . . . . . . . . . . . . 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 7815 . . . . . . . . . . . . . . . . . 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 2837 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (𝑅1β€˜πΆ)) ∧ Β¬ 𝐢 = βˆͺ 𝐢) β†’ βˆͺ 𝐢 ∈ 𝐢)
7776fvresd 6909 . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . 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 5936 . . . . . . . . . . . 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 9505 . . . . . . . . . . . 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 5874 . . . . . . . . . 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 5863 . . . . . . . . 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 2791 . . . . . . . 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 6057 . . . . . . 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 6893 . . . . . 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 4560 . . . . 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 2777 . . . 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 5248 . . 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 2773 . 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 2777 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 3475   βŠ† wss 3948  ifcif 4528  π’« cpw 4602  βˆͺ cuni 4908   ↦ cmpt 5231   E cep 5579  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679   ∘ ccom 5680  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6535   Fn wfn 6536  β€“1-1β†’wf1 6538  β€˜cfv 6541  (class class class)co 7406  recscrecs 8367   +o coa 8460   Β·o comu 8461  OrdIsocoi 9501  harchar 9548  π‘…1cr1 9754  rankcrnk 9755
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
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-reu 3378  df-rab 3434  df-v 3477  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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-oi 9502  df-r1 9756  df-rank 9757
This theorem is referenced by:  dfac12lem2  10136
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