MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem8 Structured version   Visualization version   GIF version

Theorem tfrlem8 8007
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 8001 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))}
32abeq2i 2928 . . . . . . 7 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
4 fndm 6429 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
54adantr 484 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 = 𝑧)
65eleq1d 2877 . . . . . . . . 9 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
76biimprcd 253 . . . . . . . 8 (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On))
87rexlimiv 3242 . . . . . . 7 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On)
93, 8sylbi 220 . . . . . 6 (𝑔𝐴 → dom 𝑔 ∈ On)
10 eleq1a 2888 . . . . . 6 (dom 𝑔 ∈ On → (𝑧 = dom 𝑔𝑧 ∈ On))
119, 10syl 17 . . . . 5 (𝑔𝐴 → (𝑧 = dom 𝑔𝑧 ∈ On))
1211rexlimiv 3242 . . . 4 (∃𝑔𝐴 𝑧 = dom 𝑔𝑧 ∈ On)
1312abssi 4000 . . 3 {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On
14 ssorduni 7484 . . 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 8004 . . . . 5 recs(𝐹) = 𝐴
1716dmeqi 5741 . . . 4 dom recs(𝐹) = dom 𝐴
18 dmuni 5751 . . . 4 dom 𝐴 = 𝑔𝐴 dom 𝑔
19 vex 3447 . . . . . 6 𝑔 ∈ V
2019dmex 7602 . . . . 5 dom 𝑔 ∈ V
2120dfiun2 4923 . . . 4 𝑔𝐴 dom 𝑔 = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
2217, 18, 213eqtri 2828 . . 3 dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
23 ordeq 6170 . . 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 234 1 Ord dom recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  wss 3884   cuni 4803   ciun 4884  dom cdm 5523  cres 5525  Ord word 6162  Oncon0 6163   Fn wfn 6323  cfv 6328  recscrecs 7994
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-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-tr 5140  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-pred 6120  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-wrecs 7934  df-recs 7995
This theorem is referenced by:  tfrlem10  8010  tfrlem12  8012  tfrlem13  8013  tfrlem14  8014  tfrlem15  8015  tfrlem16  8016
  Copyright terms: Public domain W3C validator