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Theorem tz7.44-3 8031
 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 6649 . . . . . 6 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
2 reseq2 5817 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
32fveq2d 6653 . . . . . 6 (𝑦 = 𝐵 → (𝐺‘(𝐹𝑦)) = (𝐺‘(𝐹𝐵)))
41, 3eqeq12d 2817 . . . . 5 (𝑦 = 𝐵 → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹𝐵) = (𝐺‘(𝐹𝐵))))
5 tz7.44.2 . . . . 5 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
64, 5vtoclga 3525 . . . 4 (𝐵𝑋 → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
76adantr 484 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐺‘(𝐹𝐵)))
82eleq1d 2877 . . . . . . 7 (𝑦 = 𝐵 → ((𝐹𝑦) ∈ V ↔ (𝐹𝐵) ∈ V))
9 tz7.44.3 . . . . . . 7 (𝑦𝑋 → (𝐹𝑦) ∈ V)
108, 9vtoclga 3525 . . . . . 6 (𝐵𝑋 → (𝐹𝐵) ∈ V)
1110adantr 484 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) ∈ V)
12 simpr 488 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → Lim 𝐵)
13 nlim0 6221 . . . . . . . . . . 11 ¬ Lim ∅
14 dmres 5844 . . . . . . . . . . . . . 14 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
15 tz7.44.5 . . . . . . . . . . . . . . . . . 18 Ord 𝑋
16 ordelss 6179 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑋𝐵𝑋) → 𝐵𝑋)
1715, 16mpan 689 . . . . . . . . . . . . . . . . 17 (𝐵𝑋𝐵𝑋)
1817adantr 484 . . . . . . . . . . . . . . . 16 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵𝑋)
19 tz7.44.4 . . . . . . . . . . . . . . . . 17 𝐹 Fn 𝑋
20 fndm 6429 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
2119, 20ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝐹 = 𝑋
2218, 21sseqtrrdi 3969 . . . . . . . . . . . . . . 15 ((𝐵𝑋 ∧ Lim 𝐵) → 𝐵 ⊆ dom 𝐹)
23 df-ss 3901 . . . . . . . . . . . . . . 15 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
2422, 23sylib 221 . . . . . . . . . . . . . 14 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐵 ∩ dom 𝐹) = 𝐵)
2514, 24syl5eq 2848 . . . . . . . . . . . . 13 ((𝐵𝑋 ∧ Lim 𝐵) → dom (𝐹𝐵) = 𝐵)
26 dmeq 5740 . . . . . . . . . . . . . 14 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = dom ∅)
27 dm0 5758 . . . . . . . . . . . . . 14 dom ∅ = ∅
2826, 27eqtrdi 2852 . . . . . . . . . . . . 13 ((𝐹𝐵) = ∅ → dom (𝐹𝐵) = ∅)
2925, 28sylan9req 2857 . . . . . . . . . . . 12 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → 𝐵 = ∅)
30 limeq 6175 . . . . . . . . . . . 12 (𝐵 = ∅ → (Lim 𝐵 ↔ Lim ∅))
3129, 30syl 17 . . . . . . . . . . 11 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → (Lim 𝐵 ↔ Lim ∅))
3213, 31mtbiri 330 . . . . . . . . . 10 (((𝐵𝑋 ∧ Lim 𝐵) ∧ (𝐹𝐵) = ∅) → ¬ Lim 𝐵)
3332ex 416 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → ((𝐹𝐵) = ∅ → ¬ Lim 𝐵))
3412, 33mt2d 138 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → ¬ (𝐹𝐵) = ∅)
3534iffalsed 4439 . . . . . . 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 6175 . . . . . . . . . 10 (dom (𝐹𝐵) = 𝐵 → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3725, 36syl 17 . . . . . . . . 9 ((𝐵𝑋 ∧ Lim 𝐵) → (Lim dom (𝐹𝐵) ↔ Lim 𝐵))
3812, 37mpbird 260 . . . . . . . 8 ((𝐵𝑋 ∧ Lim 𝐵) → Lim dom (𝐹𝐵))
3938iftrued 4436 . . . . . . 7 ((𝐵𝑋 ∧ Lim 𝐵) → if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵)))) = ran (𝐹𝐵))
4035, 39eqtrd 2836 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) = ran (𝐹𝐵))
41 rnexg 7599 . . . . . . 7 ((𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
42 uniexg 7450 . . . . . . 7 (ran (𝐹𝐵) ∈ V → ran (𝐹𝐵) ∈ V)
4311, 41, 423syl 18 . . . . . 6 ((𝐵𝑋 ∧ Lim 𝐵) → ran (𝐹𝐵) ∈ V)
4440, 43eqeltrd 2893 . . . . 5 ((𝐵𝑋 ∧ Lim 𝐵) → if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))) ∈ V)
45 eqeq1 2805 . . . . . . 7 (𝑥 = (𝐹𝐵) → (𝑥 = ∅ ↔ (𝐹𝐵) = ∅))
46 dmeq 5740 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
47 limeq 6175 . . . . . . . . 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 5774 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
5049unieqd 4817 . . . . . . . 8 (𝑥 = (𝐹𝐵) → ran 𝑥 = ran (𝐹𝐵))
51 fveq1 6648 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom 𝑥))
5246unieqd 4817 . . . . . . . . . . 11 (𝑥 = (𝐹𝐵) → dom 𝑥 = dom (𝐹𝐵))
5352fveq2d 6653 . . . . . . . . . 10 (𝑥 = (𝐹𝐵) → ((𝐹𝐵)‘ dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5451, 53eqtrd 2836 . . . . . . . . 9 (𝑥 = (𝐹𝐵) → (𝑥 dom 𝑥) = ((𝐹𝐵)‘ dom (𝐹𝐵)))
5554fveq2d 6653 . . . . . . . 8 (𝑥 = (𝐹𝐵) → (𝐻‘(𝑥 dom 𝑥)) = (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))
5648, 50, 55ifbieq12d 4455 . . . . . . 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 4453 . . . . . 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 6747 . . . . 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 587 . . . 4 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = if((𝐹𝐵) = ∅, 𝐴, if(Lim dom (𝐹𝐵), ran (𝐹𝐵), (𝐻‘((𝐹𝐵)‘ dom (𝐹𝐵))))))
6160, 40eqtrd 2836 . . 3 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐺‘(𝐹𝐵)) = ran (𝐹𝐵))
627, 61eqtrd 2836 . 2 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = ran (𝐹𝐵))
63 df-ima 5536 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
6463unieqi 4816 . 2 (𝐹𝐵) = ran (𝐹𝐵)
6562, 64eqtr4di 2854 1 ((𝐵𝑋 ∧ Lim 𝐵) → (𝐹𝐵) = (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ∩ cin 3883   ⊆ wss 3884  ∅c0 4246  ifcif 4428  ∪ cuni 4803   ↦ cmpt 5113  dom cdm 5523  ran crn 5524   ↾ cres 5525   “ cima 5526  Ord word 6162  Lim wlim 6164   Fn wfn 6323  ‘cfv 6328 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-lim 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336 This theorem is referenced by:  rdglimg  8048
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