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Theorem tz7.44-2 8447
Description: The value of 𝐹 at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypotheses
Ref Expression
tz7.44.1 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))
tz7.44.2 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
tz7.44.3 (𝑦𝑋 → (𝐹𝑦) ∈ V)
tz7.44.4 𝐹 Fn 𝑋
tz7.44.5 Ord 𝑋
Assertion
Ref Expression
tz7.44-2 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑦,𝐺   𝑥,𝐻   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐺(𝑥)   𝐻(𝑦)   𝑋(𝑥)

Proof of Theorem tz7.44-2
StepHypRef Expression
1 fveq2 6906 . . . 4 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹‘suc 𝐵))
2 reseq2 5992 . . . . 5 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹 ↾ suc 𝐵))
32fveq2d 6910 . . . 4 (𝑦 = suc 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹 ↾ suc 𝐵)))
41, 3eqeq12d 2753 . . 3 (𝑦 = suc 𝐵 → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹‘suc 𝐵) = (𝐺‘(𝐹 ↾ suc 𝐵))))
5 tz7.44.2 . . 3 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
64, 5vtoclga 3577 . 2 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐺‘(𝐹 ↾ suc 𝐵)))
7 tz7.44.1 . . 3 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))
8 eqeq1 2741 . . . 4 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ suc 𝐵) = ∅))
9 dmeq 5914 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
10 limeq 6396 . . . . . 6 (dom 𝑥 = dom (𝐹 ↾ suc 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ suc 𝐵)))
119, 10syl 17 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ suc 𝐵)))
12 rneq 5947 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
1312unieqd 4920 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
14 fveq1 6905 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom 𝑥))
159unieqd 4920 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
1615fveq2d 6910 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → ((𝐹 ↾ suc 𝐵)‘ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
1714, 16eqtrd 2777 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
1817fveq2d 6910 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))
1911, 13, 18ifbieq12d 4554 . . . 4 (𝑥 = (𝐹 ↾ suc 𝐵) → if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥))) = if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))))
208, 19ifbieq2d 4552 . . 3 (𝑥 = (𝐹 ↾ suc 𝐵) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))) = if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))))
212eleq1d 2826 . . . 4 (𝑦 = suc 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹 ↾ suc 𝐵) ∈ V))
22 tz7.44.3 . . . 4 (𝑦𝑋 → (𝐹𝑦) ∈ V)
2321, 22vtoclga 3577 . . 3 (suc 𝐵𝑋 → (𝐹 ↾ suc 𝐵) ∈ V)
24 noel 4338 . . . . . . 7 ¬ 𝐵 ∈ ∅
25 dmeq 5914 . . . . . . . . 9 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = dom ∅)
26 dm0 5931 . . . . . . . . 9 dom ∅ = ∅
2725, 26eqtrdi 2793 . . . . . . . 8 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = ∅)
28 tz7.44.5 . . . . . . . . . . . . 13 Ord 𝑋
29 ordsson 7803 . . . . . . . . . . . . 13 (Ord 𝑋𝑋 ⊆ On)
3028, 29ax-mp 5 . . . . . . . . . . . 12 𝑋 ⊆ On
31 ordtr 6398 . . . . . . . . . . . . . 14 (Ord 𝑋 → Tr 𝑋)
3228, 31ax-mp 5 . . . . . . . . . . . . 13 Tr 𝑋
33 trsuc 6471 . . . . . . . . . . . . 13 ((Tr 𝑋 ∧ suc 𝐵𝑋) → 𝐵𝑋)
3432, 33mpan 690 . . . . . . . . . . . 12 (suc 𝐵𝑋𝐵𝑋)
3530, 34sselid 3981 . . . . . . . . . . 11 (suc 𝐵𝑋𝐵 ∈ On)
36 sucidg 6465 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
3735, 36syl 17 . . . . . . . . . 10 (suc 𝐵𝑋𝐵 ∈ suc 𝐵)
38 dmres 6030 . . . . . . . . . . 11 dom (𝐹 ↾ suc 𝐵) = (suc 𝐵 ∩ dom 𝐹)
39 ordelss 6400 . . . . . . . . . . . . . 14 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵𝑋)
4028, 39mpan 690 . . . . . . . . . . . . 13 (suc 𝐵𝑋 → suc 𝐵𝑋)
41 tz7.44.4 . . . . . . . . . . . . . 14 𝐹 Fn 𝑋
4241fndmi 6672 . . . . . . . . . . . . 13 dom 𝐹 = 𝑋
4340, 42sseqtrrdi 4025 . . . . . . . . . . . 12 (suc 𝐵𝑋 → suc 𝐵 ⊆ dom 𝐹)
44 dfss2 3969 . . . . . . . . . . . 12 (suc 𝐵 ⊆ dom 𝐹 ↔ (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
4543, 44sylib 218 . . . . . . . . . . 11 (suc 𝐵𝑋 → (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
4638, 45eqtrid 2789 . . . . . . . . . 10 (suc 𝐵𝑋 → dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
4737, 46eleqtrrd 2844 . . . . . . . . 9 (suc 𝐵𝑋𝐵 ∈ dom (𝐹 ↾ suc 𝐵))
48 eleq2 2830 . . . . . . . . 9 (dom (𝐹 ↾ suc 𝐵) = ∅ → (𝐵 ∈ dom (𝐹 ↾ suc 𝐵) ↔ 𝐵 ∈ ∅))
4947, 48syl5ibcom 245 . . . . . . . 8 (suc 𝐵𝑋 → (dom (𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
5027, 49syl5 34 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
5124, 50mtoi 199 . . . . . 6 (suc 𝐵𝑋 → ¬ (𝐹 ↾ suc 𝐵) = ∅)
5251iffalsed 4536 . . . . 5 (suc 𝐵𝑋 → if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))) = if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))))
53 nlimsucg 7863 . . . . . . . 8 (𝐵 ∈ On → ¬ Lim suc 𝐵)
5435, 53syl 17 . . . . . . 7 (suc 𝐵𝑋 → ¬ Lim suc 𝐵)
55 limeq 6396 . . . . . . . 8 (dom (𝐹 ↾ suc 𝐵) = suc 𝐵 → (Lim dom (𝐹 ↾ suc 𝐵) ↔ Lim suc 𝐵))
5646, 55syl 17 . . . . . . 7 (suc 𝐵𝑋 → (Lim dom (𝐹 ↾ suc 𝐵) ↔ Lim suc 𝐵))
5754, 56mtbird 325 . . . . . 6 (suc 𝐵𝑋 → ¬ Lim dom (𝐹 ↾ suc 𝐵))
5857iffalsed 4536 . . . . 5 (suc 𝐵𝑋 → if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))) = (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))
5946unieqd 4920 . . . . . . . . 9 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
60 eloni 6394 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
61 ordunisuc 7852 . . . . . . . . . 10 (Ord 𝐵 suc 𝐵 = 𝐵)
6235, 60, 613syl 18 . . . . . . . . 9 (suc 𝐵𝑋 suc 𝐵 = 𝐵)
6359, 62eqtrd 2777 . . . . . . . 8 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = 𝐵)
6463fveq2d 6910 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = ((𝐹 ↾ suc 𝐵)‘𝐵))
6537fvresd 6926 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
6664, 65eqtrd 2777 . . . . . 6 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = (𝐹𝐵))
6766fveq2d 6910 . . . . 5 (suc 𝐵𝑋 → (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))) = (𝐻‘(𝐹𝐵)))
6852, 58, 673eqtrd 2781 . . . 4 (suc 𝐵𝑋 → if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))) = (𝐻‘(𝐹𝐵)))
69 fvex 6919 . . . 4 (𝐻‘(𝐹𝐵)) ∈ V
7068, 69eqeltrdi 2849 . . 3 (suc 𝐵𝑋 → if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))) ∈ V)
717, 20, 23, 70fvmptd3 7039 . 2 (suc 𝐵𝑋 → (𝐺‘(𝐹 ↾ suc 𝐵)) = if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))))
726, 71, 683eqtrd 2781 1 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  c0 4333  ifcif 4525   cuni 4907  cmpt 5225  Tr wtr 5259  dom cdm 5685  ran crn 5686  cres 5687  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  rdgsucg  8463
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