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Theorem tz7.44-2 8030
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 6649 . . . 4 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹‘suc 𝐵))
2 reseq2 5817 . . . . 5 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹 ↾ suc 𝐵))
32fveq2d 6653 . . . 4 (𝑦 = suc 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹 ↾ suc 𝐵)))
41, 3eqeq12d 2817 . . 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 3525 . 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 2805 . . . 4 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ suc 𝐵) = ∅))
9 dmeq 5740 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
10 limeq 6175 . . . . . 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 5774 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
1312unieqd 4817 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
14 fveq1 6648 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom 𝑥))
159unieqd 4817 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
1615fveq2d 6653 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → ((𝐹 ↾ suc 𝐵)‘ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
1714, 16eqtrd 2836 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
1817fveq2d 6653 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))
1911, 13, 18ifbieq12d 4455 . . . 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 4453 . . 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 2877 . . . 4 (𝑦 = suc 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹 ↾ suc 𝐵) ∈ V))
22 tz7.44.3 . . . 4 (𝑦𝑋 → (𝐹𝑦) ∈ V)
2321, 22vtoclga 3525 . . 3 (suc 𝐵𝑋 → (𝐹 ↾ suc 𝐵) ∈ V)
24 noel 4250 . . . . . . 7 ¬ 𝐵 ∈ ∅
25 dmeq 5740 . . . . . . . . 9 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = dom ∅)
26 dm0 5758 . . . . . . . . 9 dom ∅ = ∅
2725, 26eqtrdi 2852 . . . . . . . 8 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = ∅)
28 tz7.44.5 . . . . . . . . . . . . 13 Ord 𝑋
29 ordsson 7488 . . . . . . . . . . . . 13 (Ord 𝑋𝑋 ⊆ On)
3028, 29ax-mp 5 . . . . . . . . . . . 12 𝑋 ⊆ On
31 ordtr 6177 . . . . . . . . . . . . . 14 (Ord 𝑋 → Tr 𝑋)
3228, 31ax-mp 5 . . . . . . . . . . . . 13 Tr 𝑋
33 trsuc 6247 . . . . . . . . . . . . 13 ((Tr 𝑋 ∧ suc 𝐵𝑋) → 𝐵𝑋)
3432, 33mpan 689 . . . . . . . . . . . 12 (suc 𝐵𝑋𝐵𝑋)
3530, 34sseldi 3916 . . . . . . . . . . 11 (suc 𝐵𝑋𝐵 ∈ On)
36 sucidg 6241 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
3735, 36syl 17 . . . . . . . . . 10 (suc 𝐵𝑋𝐵 ∈ suc 𝐵)
38 dmres 5844 . . . . . . . . . . 11 dom (𝐹 ↾ suc 𝐵) = (suc 𝐵 ∩ dom 𝐹)
39 ordelss 6179 . . . . . . . . . . . . . 14 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵𝑋)
4028, 39mpan 689 . . . . . . . . . . . . 13 (suc 𝐵𝑋 → suc 𝐵𝑋)
41 tz7.44.4 . . . . . . . . . . . . . 14 𝐹 Fn 𝑋
4241fndmi 6430 . . . . . . . . . . . . 13 dom 𝐹 = 𝑋
4340, 42sseqtrrdi 3969 . . . . . . . . . . . 12 (suc 𝐵𝑋 → suc 𝐵 ⊆ dom 𝐹)
44 df-ss 3901 . . . . . . . . . . . 12 (suc 𝐵 ⊆ dom 𝐹 ↔ (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
4543, 44sylib 221 . . . . . . . . . . 11 (suc 𝐵𝑋 → (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
4638, 45syl5eq 2848 . . . . . . . . . 10 (suc 𝐵𝑋 → dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
4737, 46eleqtrrd 2896 . . . . . . . . 9 (suc 𝐵𝑋𝐵 ∈ dom (𝐹 ↾ suc 𝐵))
48 eleq2 2881 . . . . . . . . 9 (dom (𝐹 ↾ suc 𝐵) = ∅ → (𝐵 ∈ dom (𝐹 ↾ suc 𝐵) ↔ 𝐵 ∈ ∅))
4947, 48syl5ibcom 248 . . . . . . . 8 (suc 𝐵𝑋 → (dom (𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
5027, 49syl5 34 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
5124, 50mtoi 202 . . . . . 6 (suc 𝐵𝑋 → ¬ (𝐹 ↾ suc 𝐵) = ∅)
5251iffalsed 4439 . . . . 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 7541 . . . . . . . 8 (𝐵 ∈ On → ¬ Lim suc 𝐵)
5435, 53syl 17 . . . . . . 7 (suc 𝐵𝑋 → ¬ Lim suc 𝐵)
55 limeq 6175 . . . . . . . 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 328 . . . . . 6 (suc 𝐵𝑋 → ¬ Lim dom (𝐹 ↾ suc 𝐵))
5857iffalsed 4439 . . . . 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 4817 . . . . . . . . 9 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
60 eloni 6173 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
61 ordunisuc 7531 . . . . . . . . . 10 (Ord 𝐵 suc 𝐵 = 𝐵)
6235, 60, 613syl 18 . . . . . . . . 9 (suc 𝐵𝑋 suc 𝐵 = 𝐵)
6359, 62eqtrd 2836 . . . . . . . 8 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = 𝐵)
6463fveq2d 6653 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = ((𝐹 ↾ suc 𝐵)‘𝐵))
6537fvresd 6669 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
6664, 65eqtrd 2836 . . . . . 6 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = (𝐹𝐵))
6766fveq2d 6653 . . . . 5 (suc 𝐵𝑋 → (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))) = (𝐻‘(𝐹𝐵)))
6852, 58, 673eqtrd 2840 . . . 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 6662 . . . 4 (𝐻‘(𝐹𝐵)) ∈ V
7068, 69eqeltrdi 2901 . . 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 6772 . 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 2840 1 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2112  Vcvv 3444  cin 3883  wss 3884  c0 4246  ifcif 4428   cuni 4803  cmpt 5113  Tr wtr 5139  dom cdm 5523  ran crn 5524  cres 5525  Ord word 6162  Oncon0 6163  Lim wlim 6164  suc csuc 6165   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  rdgsucg  8046
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