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Theorem tfrlem8 6097
Description: Lemma for transfinite recursion. The domain of recs is 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 6090 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))}
32abeq2i 2199 . . . . . . 7 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
4 fndm 5126 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
54adantr 271 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 = 𝑧)
65eleq1d 2157 . . . . . . . . 9 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
76biimprcd 159 . . . . . . . 8 (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On))
87rexlimiv 2484 . . . . . . 7 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On)
93, 8sylbi 120 . . . . . 6 (𝑔𝐴 → dom 𝑔 ∈ On)
10 eleq1a 2160 . . . . . 6 (dom 𝑔 ∈ On → (𝑧 = dom 𝑔𝑧 ∈ On))
119, 10syl 14 . . . . 5 (𝑔𝐴 → (𝑧 = dom 𝑔𝑧 ∈ On))
1211rexlimiv 2484 . . . 4 (∃𝑔𝐴 𝑧 = dom 𝑔𝑧 ∈ On)
1312abssi 3097 . . 3 {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On
14 ssorduni 4317 . . 3 ({𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On → Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔})
1513, 14ax-mp 7 . 2 Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
161recsfval 6094 . . . . 5 recs(𝐹) = 𝐴
1716dmeqi 4650 . . . 4 dom recs(𝐹) = dom 𝐴
18 dmuni 4659 . . . 4 dom 𝐴 = 𝑔𝐴 dom 𝑔
19 vex 2623 . . . . . 6 𝑔 ∈ V
2019dmex 4712 . . . . 5 dom 𝑔 ∈ V
2120dfiun2 3770 . . . 4 𝑔𝐴 dom 𝑔 = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
2217, 18, 213eqtri 2113 . . 3 dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
23 ordeq 4208 . . 3 (dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} → (Ord dom recs(𝐹) ↔ Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}))
2422, 23ax-mp 7 . 2 (Ord dom recs(𝐹) ↔ Ord {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔})
2515, 24mpbir 145 1 Ord dom recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439  {cab 2075  wral 2360  wrex 2361  wss 3000   cuni 3659   ciun 3736  Ord word 4198  Oncon0 4199  dom cdm 4452  cres 4454   Fn wfn 5023  cfv 5028  recscrecs 6083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-tr 3943  df-iord 4202  df-on 4204  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036  df-recs 6084
This theorem is referenced by:  tfrlemi14d  6112  tfri1dALT  6130
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