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Theorem tz7.44-2 7656
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 6332 . . . 4 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹‘suc 𝐵))
2 reseq2 5529 . . . . 5 (𝑦 = suc 𝐵 → (𝐹𝑦) = (𝐹 ↾ suc 𝐵))
32fveq2d 6336 . . . 4 (𝑦 = suc 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹 ↾ suc 𝐵)))
41, 3eqeq12d 2786 . . 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 3423 . 2 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐺‘(𝐹 ↾ suc 𝐵)))
72eleq1d 2835 . . . 4 (𝑦 = suc 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹 ↾ suc 𝐵) ∈ V))
8 tz7.44.3 . . . 4 (𝑦𝑋 → (𝐹𝑦) ∈ V)
97, 8vtoclga 3423 . . 3 (suc 𝐵𝑋 → (𝐹 ↾ suc 𝐵) ∈ V)
10 noel 4067 . . . . . . 7 ¬ 𝐵 ∈ ∅
11 dmeq 5462 . . . . . . . . 9 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = dom ∅)
12 dm0 5477 . . . . . . . . 9 dom ∅ = ∅
1311, 12syl6eq 2821 . . . . . . . 8 ((𝐹 ↾ suc 𝐵) = ∅ → dom (𝐹 ↾ suc 𝐵) = ∅)
14 tz7.44.5 . . . . . . . . . . . . 13 Ord 𝑋
15 ordsson 7136 . . . . . . . . . . . . 13 (Ord 𝑋𝑋 ⊆ On)
1614, 15ax-mp 5 . . . . . . . . . . . 12 𝑋 ⊆ On
17 ordtr 5880 . . . . . . . . . . . . . 14 (Ord 𝑋 → Tr 𝑋)
1814, 17ax-mp 5 . . . . . . . . . . . . 13 Tr 𝑋
19 trsuc 5953 . . . . . . . . . . . . 13 ((Tr 𝑋 ∧ suc 𝐵𝑋) → 𝐵𝑋)
2018, 19mpan 670 . . . . . . . . . . . 12 (suc 𝐵𝑋𝐵𝑋)
2116, 20sseldi 3750 . . . . . . . . . . 11 (suc 𝐵𝑋𝐵 ∈ On)
22 sucidg 5946 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
2321, 22syl 17 . . . . . . . . . 10 (suc 𝐵𝑋𝐵 ∈ suc 𝐵)
24 dmres 5560 . . . . . . . . . . 11 dom (𝐹 ↾ suc 𝐵) = (suc 𝐵 ∩ dom 𝐹)
25 ordelss 5882 . . . . . . . . . . . . . 14 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵𝑋)
2614, 25mpan 670 . . . . . . . . . . . . 13 (suc 𝐵𝑋 → suc 𝐵𝑋)
27 tz7.44.4 . . . . . . . . . . . . . 14 𝐹 Fn 𝑋
28 fndm 6130 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
2927, 28ax-mp 5 . . . . . . . . . . . . 13 dom 𝐹 = 𝑋
3026, 29syl6sseqr 3801 . . . . . . . . . . . 12 (suc 𝐵𝑋 → suc 𝐵 ⊆ dom 𝐹)
31 df-ss 3737 . . . . . . . . . . . 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 2817 . . . . . . . . . 10 (suc 𝐵𝑋 → dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
3423, 33eleqtrrd 2853 . . . . . . . . 9 (suc 𝐵𝑋𝐵 ∈ dom (𝐹 ↾ suc 𝐵))
35 eleq2 2839 . . . . . . . . 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 4236 . . . . 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 7189 . . . . . . . 8 (𝐵 ∈ On → ¬ Lim suc 𝐵)
4121, 40syl 17 . . . . . . 7 (suc 𝐵𝑋 → ¬ Lim suc 𝐵)
42 limeq 5878 . . . . . . . 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 4236 . . . . 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 4584 . . . . . . . . 9 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = suc 𝐵)
47 eloni 5876 . . . . . . . . . . 11 (𝐵 ∈ On → Ord 𝐵)
48 ordunisuc 7179 . . . . . . . . . . 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 2805 . . . . . . . 8 (suc 𝐵𝑋 dom (𝐹 ↾ suc 𝐵) = 𝐵)
5251fveq2d 6336 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = ((𝐹 ↾ suc 𝐵)‘𝐵))
53 fvres 6348 . . . . . . . 8 (𝐵 ∈ suc 𝐵 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
5423, 53syl 17 . . . . . . 7 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘𝐵) = (𝐹𝐵))
5552, 54eqtrd 2805 . . . . . 6 (suc 𝐵𝑋 → ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)) = (𝐹𝐵))
5655fveq2d 6336 . . . . 5 (suc 𝐵𝑋 → (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))) = (𝐻‘(𝐹𝐵)))
5739, 45, 563eqtrd 2809 . . . 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 6342 . . . 4 (𝐻‘(𝐹𝐵)) ∈ V
5957, 58syl6eqel 2858 . . 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 2775 . . . . 5 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 = ∅ ↔ (𝐹 ↾ suc 𝐵) = ∅))
61 dmeq 5462 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
62 limeq 5878 . . . . . . 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 5489 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
6564unieqd 4584 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → ran 𝑥 = ran (𝐹 ↾ suc 𝐵))
66 fveq1 6331 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom 𝑥))
6761unieqd 4584 . . . . . . . . 9 (𝑥 = (𝐹 ↾ suc 𝐵) → dom 𝑥 = dom (𝐹 ↾ suc 𝐵))
6867fveq2d 6336 . . . . . . . 8 (𝑥 = (𝐹 ↾ suc 𝐵) → ((𝐹 ↾ suc 𝐵)‘ dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
6966, 68eqtrd 2805 . . . . . . 7 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝑥 dom 𝑥) = ((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵)))
7069fveq2d 6336 . . . . . 6 (𝑥 = (𝐹 ↾ suc 𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹 ↾ suc 𝐵)‘ dom (𝐹 ↾ suc 𝐵))))
7163, 65, 70ifbieq12d 4252 . . . . 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 4250 . . . 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 6422 . . 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 573 . 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 2809 1 (suc 𝐵𝑋 → (𝐹‘suc 𝐵) = (𝐻‘(𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  wss 3723  c0 4063  ifcif 4225   cuni 4574  cmpt 4863  Tr wtr 4886  dom cdm 5249  ran crn 5250  cres 5251  Ord word 5865  Oncon0 5866  Lim wlim 5867  suc csuc 5868   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  rdgsucg  7672
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