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Theorem tz7.44-2 7548
 Description: The value of 𝐹 at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (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 6229 . . . 4 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹‘suc 𝐵))
2 reseq2 5423 . . . . 5 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹 ↾ suc 𝐵))
32fveq2d 6233 . . . 4 (𝑦 = suc 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹 ↾ suc 𝐵)))
41, 3eqeq12d 2666 . . 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 3303 . 2 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐺‘(𝐹 ↾ suc 𝐵)))
72eleq1d 2715 . . . 4 (𝑦 = suc 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹 ↾ suc 𝐵) ∈ V))
8 tz7.44.3 . . . 4 (𝑦𝑋 → (𝐹𝑦) ∈ V)
97, 8vtoclga 3303 . . 3 (suc 𝐵𝑋 → (𝐹 ↾ suc 𝐵) ∈ V)
10 noel 3952 . . . . . . 7 ¬ 𝐵 ∈ ∅
11 dmeq 5356 . . . . . . . . 9 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = dom ∅)
12 dm0 5371 . . . . . . . . 9 dom ∅ = ∅
1311, 12syl6eq 2701 . . . . . . . 8 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = ∅)
14 tz7.44.5 . . . . . . . . . . . . 13 Ord 𝑋
15 ordsson 7031 . . . . . . . . . . . . 13 (Ord 𝑋𝑋 ⊆ On)
1614, 15ax-mp 5 . . . . . . . . . . . 12 𝑋 ⊆ On
17 ordtr 5775 . . . . . . . . . . . . . 14 (Ord 𝑋 → Tr 𝑋)
1814, 17ax-mp 5 . . . . . . . . . . . . 13 Tr 𝑋
19 trsuc 5848 . . . . . . . . . . . . 13 ((Tr 𝑋 ∧ suc 𝐵𝑋) → 𝐵𝑋)
2018, 19mpan 706 . . . . . . . . . . . 12 (suc 𝐵𝑋𝐵𝑋)
2116, 20sseldi 3634 . . . . . . . . . . 11 (suc 𝐵𝑋𝐵 ∈ On)
22 sucidg 5841 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
2321, 22syl 17 . . . . . . . . . 10 (suc 𝐵𝑋𝐵 ∈ suc 𝐵)
24 dmres 5454 . . . . . . . . . . 11 dom (𝐹 ↾ suc 𝐵) = (suc 𝐵 ∩ dom 𝐹)
25 ordelss 5777 . . . . . . . . . . . . . 14 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵𝑋)
2614, 25mpan 706 . . . . . . . . . . . . 13 (suc 𝐵𝑋 → suc 𝐵𝑋)
27 tz7.44.4 . . . . . . . . . . . . . 14 𝐹 Fn 𝑋
28 fndm 6028 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
2927, 28ax-mp 5 . . . . . . . . . . . . 13 dom 𝐹 = 𝑋
3026, 29syl6sseqr 3685 . . . . . . . . . . . 12 (suc 𝐵𝑋 → suc 𝐵 ⊆ dom 𝐹)
31 df-ss 3621 . . . . . . . . . . . 12 (suc 𝐵 ⊆ dom 𝐹 ↔ (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
3230, 31sylib 208 . . . . . . . . . . 11 (suc 𝐵𝑋 → (suc 𝐵 ∩ dom 𝐹) = suc 𝐵)
3324, 32syl5eq 2697 . . . . . . . . . 10 (suc 𝐵𝑋 → dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
3423, 33eleqtrrd 2733 . . . . . . . . 9 (suc 𝐵𝑋𝐵 ∈ dom (𝐹 ↾ suc 𝐵))
35 eleq2 2719 . . . . . . . . 9 (dom (𝐹 ↾ suc 𝐵) = ∅ → (𝐵 ∈ dom (𝐹 ↾ suc 𝐵) ↔ 𝐵 ∈ ∅))
3634, 35syl5ibcom 235 . . . . . . . 8 (suc 𝐵𝑋 → (dom (𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
3713, 36syl5 34 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵) = ∅ → 𝐵 ∈ ∅))
3810, 37mtoi 190 . . . . . 6 (suc 𝐵𝑋 → ¬ (𝐹 ↾ suc 𝐵) = ∅)
3938iffalsed 4130 . . . . 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)))))
40 nlimsucg 7084 . . . . . . . 8 (𝐵 ∈ On → ¬ Lim suc 𝐵)
4121, 40syl 17 . . . . . . 7 (suc 𝐵𝑋 → ¬ Lim suc 𝐵)
42 limeq 5773 . . . . . . . 8 (dom (𝐹 ↾ suc 𝐵) = suc 𝐵 → (Lim dom (𝐹 ↾ suc 𝐵) ↔ Lim suc 𝐵))
4333, 42syl 17 . . . . . . 7 (suc 𝐵𝑋 → (Lim dom (𝐹 ↾ suc 𝐵) ↔ Lim suc 𝐵))
4441, 43mtbird 314 . . . . . 6 (suc 𝐵𝑋 → ¬ Lim dom (𝐹 ↾ suc 𝐵))
4544iffalsed 4130 . . . . 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))))
4633unieqd 4478 . . . . . . . . 9 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
47 eloni 5771 . . . . . . . . . . 11 (𝐵 ∈ On → Ord 𝐵)
48 ordunisuc 7074 . . . . . . . . . . 11 (Ord 𝐵 suc 𝐵 = 𝐵)
4947, 48syl 17 . . . . . . . . . 10 (𝐵 ∈ On → suc 𝐵 = 𝐵)
5021, 49syl 17 . . . . . . . . 9 (suc 𝐵𝑋 suc 𝐵 = 𝐵)
5146, 50eqtrd 2685 . . . . . . . 8 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = 𝐵)
5251fveq2d 6233 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = ((𝐹 ↾ suc 𝐵)‘𝐵))
53 fvres 6245 . . . . . . . 8 (𝐵 ∈ suc 𝐵 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
5423, 53syl 17 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
5552, 54eqtrd 2685 . . . . . 6 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = (𝐹𝐵))
5655fveq2d 6233 . . . . 5 (suc 𝐵𝑋 → (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))) = (𝐻‘(𝐹𝐵)))
5739, 45, 563eqtrd 2689 . . . 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)))
58 fvex 6239 . . . 4 (𝐻‘(𝐹𝐵)) ∈ V
5957, 58syl6eqel 2738 . . 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)
60 eqeq1 2655 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ suc 𝐵) = ∅))
61 dmeq 5356 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
62 limeq 5773 . . . . . . 7 (dom 𝑥 = dom (𝐹 ↾ suc 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ suc 𝐵)))
6361, 62syl 17 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹 ↾ suc 𝐵)))
64 rneq 5383 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
6564unieqd 4478 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
66 fveq1 6228 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom 𝑥))
6761unieqd 4478 . . . . . . . . 9 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
6867fveq2d 6233 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → ((𝐹 ↾ suc 𝐵)‘ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
6966, 68eqtrd 2685 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
7069fveq2d 6233 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))
7163, 65, 70ifbieq12d 4146 . . . . 5 (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)))))
7260, 71ifbieq2d 4144 . . . 4 (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))))))
73 tz7.44.1 . . . 4 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))
7472, 73fvmptg 6319 . . 3 (((𝐹 ↾ suc 𝐵) ∈ V ∧ if((𝐹 ↾ suc 𝐵) = ∅, 𝐴, if(Lim dom (𝐹 ↾ suc 𝐵), ran (𝐹 ↾ suc 𝐵), (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))) ∈ V) → (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))))))
759, 59, 74syl2anc 694 . 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))))))
766, 75, 573eqtrd 2689 1 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ifcif 4119  ∪ cuni 4468   ↦ cmpt 4762  Tr wtr 4785  dom cdm 5143  ran crn 5144   ↾ cres 5145  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by:  rdgsucg  7564
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