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Theorem tz7.44-3 8408
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 6892 . . . . . 6 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2 reseq2 5977 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
32fveq2d 6896 . . . . . 6 (𝑦 = 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹𝐵)))
41, 3eqeq12d 2749 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹𝐵) = (𝐺‘(𝐹𝐵))))
5 tz7.44.2 . . . . 5 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
64, 5vtoclga 3566 . . . 4 (𝐵𝑋 → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
76adantr 482 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
82eleq1d 2819 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹𝐵) ∈ V))
9 tz7.44.3 . . . . . . 7 (𝑦𝑋 → (𝐹𝑦) ∈ V)
108, 9vtoclga 3566 . . . . . 6 (𝐵𝑋 → (𝐹𝐵) ∈ V)
1110adantr 482 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) ∈ V)
12 simpr 486 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → Lim 𝐵)
13 nlim0 6424 . . . . . . . . . . 11 ¬ Lim ∅
14 dmres 6004 . . . . . . . . . . . . . 14 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
15 tz7.44.5 . . . . . . . . . . . . . . . . . 18 Ord 𝑋
16 ordelss 6381 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑋𝐵𝑋) → 𝐵𝑋)
1715, 16mpan 689 . . . . . . . . . . . . . . . . 17 (𝐵𝑋𝐵𝑋)
1817adantr 482 . . . . . . . . . . . . . . . 16 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵𝑋)
19 tz7.44.4 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
20 fndm 6653 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝐹 = 𝑋
2218, 21sseqtrrdi 4034 . . . . . . . . . . . . . . 15 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵 ⊆ dom 𝐹)
23 df-ss 3966 . . . . . . . . . . . . . . 15 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
2422, 23sylib 217 . . . . . . . . . . . . . 14 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐵 ∩ dom 𝐹) = 𝐵)
2514, 24eqtrid 2785 . . . . . . . . . . . . 13 ((𝐵𝑋 ∧ Lim 𝐵) → dom (𝐹𝐵) = 𝐵)
26 dmeq 5904 . . . . . . . . . . . . . 14 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = dom ∅)
27 dm0 5921 . . . . . . . . . . . . . 14 dom ∅ = ∅
2826, 27eqtrdi 2789 . . . . . . . . . . . . 13 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = ∅)
2925, 28sylan9req 2794 . . . . . . . . . . . 12 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → 𝐵 = ∅)
30 limeq 6377 . . . . . . . . . . . 12 (𝐵 = ∅ → (Lim 𝐵 ↔ Lim ∅))
3129, 30syl 17 . . . . . . . . . . 11 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → (Lim 𝐵 ↔ Lim ∅))
3213, 31mtbiri 327 . . . . . . . . . 10 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → ¬ Lim 𝐵)
3332ex 414 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → ((𝐹𝐵) = ∅ → ¬ Lim 𝐵))
3412, 33mt2d 136 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → ¬ (𝐹𝐵) = ∅)
3534iffalsed 4540 . . . . . . 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 6377 . . . . . . . . . 10 (dom (𝐹𝐵) = 𝐵 → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3725, 36syl 17 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3812, 37mpbird 257 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → Lim dom (𝐹𝐵))
3938iftrued 4537 . . . . . . 7 ((𝐵𝑋 ∧ Lim 𝐵) → if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵)))) = ran (𝐹𝐵))
4035, 39eqtrd 2773 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) = ran (𝐹𝐵))
41 rnexg 7895 . . . . . . 7 ((𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
42 uniexg 7730 . . . . . . 7 (ran (𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
4311, 41, 423syl 18 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → ran (𝐹𝐵) ∈ V)
4440, 43eqeltrd 2834 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) ∈ V)
45 eqeq1 2737 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝑥 = ∅ ↔ (𝐹𝐵) = ∅))
46 dmeq 5904 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
47 limeq 6377 . . . . . . . . 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 5936 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
5049unieqd 4923 . . . . . . . 8 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
51 fveq1 6891 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom 𝑥))
5246unieqd 4923 . . . . . . . . . . 11 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
5352fveq2d 6896 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → ((𝐹𝐵)‘ dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5451, 53eqtrd 2773 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5554fveq2d 6896 . . . . . . . 8 (𝑥 = (𝐹𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))
5648, 50, 55ifbieq12d 4557 . . . . . . 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 4555 . . . . . 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 6997 . . . . 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 585 . . . 4 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))))
6160, 40eqtrd 2773 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = ran (𝐹𝐵))
627, 61eqtrd 2773 . 2 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = ran (𝐹𝐵))
63 df-ima 5690 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
6463unieqi 4922 . 2 (𝐹𝐵) = ran (𝐹𝐵)
6562, 64eqtr4di 2791 1 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cin 3948  wss 3949  c0 4323  ifcif 4529   cuni 4909  cmpt 5232  dom cdm 5677  ran crn 5678  cres 5679  cima 5680  Ord word 6364  Lim wlim 6366   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-lim 6370  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  rdglimg  8425
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