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Theorem tz7.44-3 8346
Description: The value of 𝐹 at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
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
tz7.44-3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (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 tz7.44-3
StepHypRef Expression
1 fveq2 6839 . . . . . 6 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2 reseq2 5930 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
32fveq2d 6843 . . . . . 6 (𝑦 = 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹𝐵)))
41, 3eqeq12d 2753 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹𝐵) = (𝐺‘(𝐹𝐵))))
5 tz7.44.2 . . . . 5 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
64, 5vtoclga 3532 . . . 4 (𝐵𝑋 → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
76adantr 481 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
82eleq1d 2822 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹𝐵) ∈ V))
9 tz7.44.3 . . . . . . 7 (𝑦𝑋 → (𝐹𝑦) ∈ V)
108, 9vtoclga 3532 . . . . . 6 (𝐵𝑋 → (𝐹𝐵) ∈ V)
1110adantr 481 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) ∈ V)
12 simpr 485 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → Lim 𝐵)
13 nlim0 6374 . . . . . . . . . . 11 ¬ Lim ∅
14 dmres 5957 . . . . . . . . . . . . . 14 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
15 tz7.44.5 . . . . . . . . . . . . . . . . . 18 Ord 𝑋
16 ordelss 6331 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑋𝐵𝑋) → 𝐵𝑋)
1715, 16mpan 688 . . . . . . . . . . . . . . . . 17 (𝐵𝑋𝐵𝑋)
1817adantr 481 . . . . . . . . . . . . . . . 16 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵𝑋)
19 tz7.44.4 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
20 fndm 6602 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝐹 = 𝑋
2218, 21sseqtrrdi 3993 . . . . . . . . . . . . . . 15 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵 ⊆ dom 𝐹)
23 df-ss 3925 . . . . . . . . . . . . . . 15 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
2422, 23sylib 217 . . . . . . . . . . . . . 14 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐵 ∩ dom 𝐹) = 𝐵)
2514, 24eqtrid 2789 . . . . . . . . . . . . 13 ((𝐵𝑋 ∧ Lim 𝐵) → dom (𝐹𝐵) = 𝐵)
26 dmeq 5857 . . . . . . . . . . . . . 14 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = dom ∅)
27 dm0 5874 . . . . . . . . . . . . . 14 dom ∅ = ∅
2826, 27eqtrdi 2793 . . . . . . . . . . . . 13 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = ∅)
2925, 28sylan9req 2798 . . . . . . . . . . . 12 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → 𝐵 = ∅)
30 limeq 6327 . . . . . . . . . . . 12 (𝐵 = ∅ → (Lim 𝐵 ↔ Lim ∅))
3129, 30syl 17 . . . . . . . . . . 11 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → (Lim 𝐵 ↔ Lim ∅))
3213, 31mtbiri 326 . . . . . . . . . 10 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → ¬ Lim 𝐵)
3332ex 413 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → ((𝐹𝐵) = ∅ → ¬ Lim 𝐵))
3412, 33mt2d 136 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → ¬ (𝐹𝐵) = ∅)
3534iffalsed 4495 . . . . . . 7 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) = if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵)))))
36 limeq 6327 . . . . . . . . . 10 (dom (𝐹𝐵) = 𝐵 → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3725, 36syl 17 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3812, 37mpbird 256 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → Lim dom (𝐹𝐵))
3938iftrued 4492 . . . . . . 7 ((𝐵𝑋 ∧ Lim 𝐵) → if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵)))) = ran (𝐹𝐵))
4035, 39eqtrd 2777 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) = ran (𝐹𝐵))
41 rnexg 7833 . . . . . . 7 ((𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
42 uniexg 7669 . . . . . . 7 (ran (𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
4311, 41, 423syl 18 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → ran (𝐹𝐵) ∈ V)
4440, 43eqeltrd 2838 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) ∈ V)
45 eqeq1 2741 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝑥 = ∅ ↔ (𝐹𝐵) = ∅))
46 dmeq 5857 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
47 limeq 6327 . . . . . . . . 9 (dom 𝑥 = dom (𝐹𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹𝐵)))
4846, 47syl 17 . . . . . . . 8 (𝑥 = (𝐹𝐵) → (Lim dom 𝑥 ↔ Lim dom (𝐹𝐵)))
49 rneq 5889 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
5049unieqd 4877 . . . . . . . 8 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
51 fveq1 6838 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom 𝑥))
5246unieqd 4877 . . . . . . . . . . 11 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
5352fveq2d 6843 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → ((𝐹𝐵)‘ dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5451, 53eqtrd 2777 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5554fveq2d 6843 . . . . . . . 8 (𝑥 = (𝐹𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))
5648, 50, 55ifbieq12d 4512 . . . . . . 7 (𝑥 = (𝐹𝐵) → if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥))) = if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵)))))
5745, 56ifbieq2d 4510 . . . . . 6 (𝑥 = (𝐹𝐵) → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))) = if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))))
58 tz7.44.1 . . . . . 6 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))
5957, 58fvmptg 6943 . . . . 5 (((𝐹𝐵) ∈ V ∧ if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) ∈ V) → (𝐺‘(𝐹𝐵)) = if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))))
6011, 44, 59syl2anc 584 . . . 4 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))))
6160, 40eqtrd 2777 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = ran (𝐹𝐵))
627, 61eqtrd 2777 . 2 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = ran (𝐹𝐵))
63 df-ima 5644 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
6463unieqi 4876 . 2 (𝐹𝐵) = ran (𝐹𝐵)
6562, 64eqtr4di 2795 1 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  cin 3907  wss 3908  c0 4280  ifcif 4484   cuni 4863  cmpt 5186  dom cdm 5631  ran crn 5632  cres 5633  cima 5634  Ord word 6314  Lim wlim 6316   Fn wfn 6488  cfv 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6318  df-lim 6320  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501
This theorem is referenced by:  rdglimg  8363
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