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Theorem tz7.44-3 7489
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 6178 . . . . . 6 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2 reseq2 5380 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
32fveq2d 6182 . . . . . 6 (𝑦 = 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹𝐵)))
41, 3eqeq12d 2635 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹𝐵) = (𝐺‘(𝐹𝐵))))
5 tz7.44.2 . . . . 5 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
64, 5vtoclga 3267 . . . 4 (𝐵𝑋 → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
76adantr 481 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
82eleq1d 2684 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹𝐵) ∈ V))
9 tz7.44.3 . . . . . . 7 (𝑦𝑋 → (𝐹𝑦) ∈ V)
108, 9vtoclga 3267 . . . . . 6 (𝐵𝑋 → (𝐹𝐵) ∈ V)
1110adantr 481 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) ∈ V)
12 simpr 477 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → Lim 𝐵)
13 nlim0 5771 . . . . . . . . . . 11 ¬ Lim ∅
14 dmres 5407 . . . . . . . . . . . . . 14 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
15 tz7.44.5 . . . . . . . . . . . . . . . . . 18 Ord 𝑋
16 ordelss 5727 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑋𝐵𝑋) → 𝐵𝑋)
1715, 16mpan 705 . . . . . . . . . . . . . . . . 17 (𝐵𝑋𝐵𝑋)
1817adantr 481 . . . . . . . . . . . . . . . 16 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵𝑋)
19 tz7.44.4 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
20 fndm 5978 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝐹 = 𝑋
2218, 21syl6sseqr 3644 . . . . . . . . . . . . . . 15 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵 ⊆ dom 𝐹)
23 df-ss 3581 . . . . . . . . . . . . . . 15 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
2422, 23sylib 208 . . . . . . . . . . . . . 14 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐵 ∩ dom 𝐹) = 𝐵)
2514, 24syl5eq 2666 . . . . . . . . . . . . 13 ((𝐵𝑋 ∧ Lim 𝐵) → dom (𝐹𝐵) = 𝐵)
26 dmeq 5313 . . . . . . . . . . . . . 14 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = dom ∅)
27 dm0 5328 . . . . . . . . . . . . . 14 dom ∅ = ∅
2826, 27syl6eq 2670 . . . . . . . . . . . . 13 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = ∅)
2925, 28sylan9req 2675 . . . . . . . . . . . 12 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → 𝐵 = ∅)
30 limeq 5723 . . . . . . . . . . . 12 (𝐵 = ∅ → (Lim 𝐵 ↔ Lim ∅))
3129, 30syl 17 . . . . . . . . . . 11 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → (Lim 𝐵 ↔ Lim ∅))
3213, 31mtbiri 317 . . . . . . . . . 10 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → ¬ Lim 𝐵)
3332ex 450 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → ((𝐹𝐵) = ∅ → ¬ Lim 𝐵))
3412, 33mt2d 131 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → ¬ (𝐹𝐵) = ∅)
3534iffalsed 4088 . . . . . . 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 5723 . . . . . . . . . 10 (dom (𝐹𝐵) = 𝐵 → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3725, 36syl 17 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3812, 37mpbird 247 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → Lim dom (𝐹𝐵))
3938iftrued 4085 . . . . . . 7 ((𝐵𝑋 ∧ Lim 𝐵) → if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵)))) = ran (𝐹𝐵))
4035, 39eqtrd 2654 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) = ran (𝐹𝐵))
41 rnexg 7083 . . . . . . 7 ((𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
42 uniexg 6940 . . . . . . 7 (ran (𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
4311, 41, 423syl 18 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → ran (𝐹𝐵) ∈ V)
4440, 43eqeltrd 2699 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) ∈ V)
45 eqeq1 2624 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝑥 = ∅ ↔ (𝐹𝐵) = ∅))
46 dmeq 5313 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
47 limeq 5723 . . . . . . . . 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 5340 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
5049unieqd 4437 . . . . . . . 8 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
51 fveq1 6177 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom 𝑥))
5246unieqd 4437 . . . . . . . . . . 11 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
5352fveq2d 6182 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → ((𝐹𝐵)‘ dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5451, 53eqtrd 2654 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5554fveq2d 6182 . . . . . . . 8 (𝑥 = (𝐹𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))
5648, 50, 55ifbieq12d 4104 . . . . . . 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 4102 . . . . . 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 6267 . . . . 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 692 . . . 4 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))))
6160, 40eqtrd 2654 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = ran (𝐹𝐵))
627, 61eqtrd 2654 . 2 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = ran (𝐹𝐵))
63 df-ima 5117 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
6463unieqi 4436 . 2 (𝐹𝐵) = ran (𝐹𝐵)
6562, 64syl6eqr 2672 1 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  wss 3567  c0 3907  ifcif 4077   cuni 4427  cmpt 4720  dom cdm 5104  ran crn 5105  cres 5106  cima 5107  Ord word 5710  Lim wlim 5712   Fn wfn 5871  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-lim 5716  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  rdglimg  7506
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