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Theorem tfrlem8 8405
Description: Lemma for transfinite recursion. The domain of recs is an ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem8 Ord dom recs(𝐹)
Distinct variable group:   𝑥,𝑓,𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem8
Dummy variables 𝑔 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem3 8399 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))}
32eqabri 2869 . . . . . . 7 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
4 fndm 6658 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
54adantr 479 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 = 𝑧)
65eleq1d 2810 . . . . . . . . 9 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
76biimprcd 249 . . . . . . . 8 (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On))
87rexlimiv 3137 . . . . . . 7 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On)
93, 8sylbi 216 . . . . . 6 (𝑔𝐴 → dom 𝑔 ∈ On)
10 eleq1a 2820 . . . . . 6 (dom 𝑔 ∈ On → (𝑧 = dom 𝑔𝑧 ∈ On))
119, 10syl 17 . . . . 5 (𝑔𝐴 → (𝑧 = dom 𝑔𝑧 ∈ On))
1211rexlimiv 3137 . . . 4 (∃𝑔𝐴 𝑧 = dom 𝑔𝑧 ∈ On)
1312abssi 4063 . . 3 {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On
14 ssorduni 7782 . . 3 ({𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On → Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔})
1513, 14ax-mp 5 . 2 Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
161recsfval 8402 . . . . 5 recs(𝐹) = 𝐴
1716dmeqi 5907 . . . 4 dom recs(𝐹) = dom 𝐴
18 dmuni 5917 . . . 4 dom 𝐴 = 𝑔𝐴 dom 𝑔
19 vex 3465 . . . . . 6 𝑔 ∈ V
2019dmex 7917 . . . . 5 dom 𝑔 ∈ V
2120dfiun2 5037 . . . 4 𝑔𝐴 dom 𝑔 = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
2217, 18, 213eqtri 2757 . . 3 dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
23 ordeq 6378 . . 3 (dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} → (Ord dom recs(𝐹) ↔ Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}))
2422, 23ax-mp 5 . 2 (Ord dom recs(𝐹) ↔ Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔})
2515, 24mpbir 230 1 Ord dom recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2702  wral 3050  wrex 3059  wss 3944   cuni 4909   ciun 4997  dom cdm 5678  cres 5680  Ord word 6370  Oncon0 6371   Fn wfn 6544  cfv 6549  recscrecs 8391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fo 6555  df-fv 6557  df-ov 7422  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392
This theorem is referenced by:  tfrlem10  8408  tfrlem12  8410  tfrlem13  8411  tfrlem14  8412  tfrlem15  8413  tfrlem16  8414
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