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Theorem tz7.44-2 8446
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 6907 . . . 4 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹‘suc 𝐵))
2 reseq2 5995 . . . . 5 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹 ↾ suc 𝐵))
32fveq2d 6911 . . . 4 (𝑦 = suc 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹 ↾ suc 𝐵)))
41, 3eqeq12d 2751 . . 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 2739 . . . 4 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ suc 𝐵) = ∅))
9 dmeq 5917 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
10 limeq 6398 . . . . . 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 5950 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
1312unieqd 4925 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
14 fveq1 6906 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom 𝑥))
159unieqd 4925 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
1615fveq2d 6911 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → ((𝐹 ↾ suc 𝐵)‘ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
1714, 16eqtrd 2775 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
1817fveq2d 6911 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))
1911, 13, 18ifbieq12d 4559 . . . 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 4557 . . 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 2824 . . . 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 4344 . . . . . . 7 ¬ 𝐵 ∈ ∅
25 dmeq 5917 . . . . . . . . 9 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = dom ∅)
26 dm0 5934 . . . . . . . . 9 dom ∅ = ∅
2725, 26eqtrdi 2791 . . . . . . . 8 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = ∅)
28 tz7.44.5 . . . . . . . . . . . . 13 Ord 𝑋
29 ordsson 7802 . . . . . . . . . . . . 13 (Ord 𝑋𝑋 ⊆ On)
3028, 29ax-mp 5 . . . . . . . . . . . 12 𝑋 ⊆ On
31 ordtr 6400 . . . . . . . . . . . . . 14 (Ord 𝑋 → Tr 𝑋)
3228, 31ax-mp 5 . . . . . . . . . . . . 13 Tr 𝑋
33 trsuc 6473 . . . . . . . . . . . . 13 ((Tr 𝑋 ∧ suc 𝐵𝑋) → 𝐵𝑋)
3432, 33mpan 690 . . . . . . . . . . . 12 (suc 𝐵𝑋𝐵𝑋)
3530, 34sselid 3993 . . . . . . . . . . 11 (suc 𝐵𝑋𝐵 ∈ On)
36 sucidg 6467 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
3735, 36syl 17 . . . . . . . . . 10 (suc 𝐵𝑋𝐵 ∈ suc 𝐵)
38 dmres 6032 . . . . . . . . . . 11 dom (𝐹 ↾ suc 𝐵) = (suc 𝐵 ∩ dom 𝐹)
39 ordelss 6402 . . . . . . . . . . . . . 14 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵𝑋)
4028, 39mpan 690 . . . . . . . . . . . . 13 (suc 𝐵𝑋 → suc 𝐵𝑋)
41 tz7.44.4 . . . . . . . . . . . . . 14 𝐹 Fn 𝑋
4241fndmi 6673 . . . . . . . . . . . . 13 dom 𝐹 = 𝑋
4340, 42sseqtrrdi 4047 . . . . . . . . . . . 12 (suc 𝐵𝑋 → suc 𝐵 ⊆ dom 𝐹)
44 dfss2 3981 . . . . . . . . . . . 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 2787 . . . . . . . . . 10 (suc 𝐵𝑋 → dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
4737, 46eleqtrrd 2842 . . . . . . . . 9 (suc 𝐵𝑋𝐵 ∈ dom (𝐹 ↾ suc 𝐵))
48 eleq2 2828 . . . . . . . . 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 4542 . . . . 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 6398 . . . . . . . 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 4542 . . . . 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 4925 . . . . . . . . 9 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
60 eloni 6396 . . . . . . . . . 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 2775 . . . . . . . 8 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = 𝐵)
6463fveq2d 6911 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = ((𝐹 ↾ suc 𝐵)‘𝐵))
6537fvresd 6927 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
6664, 65eqtrd 2775 . . . . . 6 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = (𝐹𝐵))
6766fveq2d 6911 . . . . 5 (suc 𝐵𝑋 → (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))) = (𝐻‘(𝐹𝐵)))
6852, 58, 673eqtrd 2779 . . . 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 6920 . . . 4 (𝐻‘(𝐹𝐵)) ∈ V
7068, 69eqeltrdi 2847 . . 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 2779 1 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  c0 4339  ifcif 4531   cuni 4912  cmpt 5231  Tr wtr 5265  dom cdm 5689  ran crn 5690  cres 5691  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  rdgsucg  8462
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