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Theorem tfrlem8 6297
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 6290 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))}
32abeq2i 2281 . . . . . . 7 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
4 fndm 5297 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
54adantr 274 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 = 𝑧)
65eleq1d 2239 . . . . . . . . 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 2581 . . . . . . 7 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On)
93, 8sylbi 120 . . . . . 6 (𝑔𝐴 → dom 𝑔 ∈ On)
10 eleq1a 2242 . . . . . 6 (dom 𝑔 ∈ On → (𝑧 = dom 𝑔𝑧 ∈ On))
119, 10syl 14 . . . . 5 (𝑔𝐴 → (𝑧 = dom 𝑔𝑧 ∈ On))
1211rexlimiv 2581 . . . 4 (∃𝑔𝐴 𝑧 = dom 𝑔𝑧 ∈ On)
1312abssi 3222 . . 3 {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On
14 ssorduni 4471 . . 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 6294 . . . . 5 recs(𝐹) = 𝐴
1716dmeqi 4812 . . . 4 dom recs(𝐹) = dom 𝐴
18 dmuni 4821 . . . 4 dom 𝐴 = 𝑔𝐴 dom 𝑔
19 vex 2733 . . . . . 6 𝑔 ∈ V
2019dmex 4877 . . . . 5 dom 𝑔 ∈ V
2120dfiun2 3907 . . . 4 𝑔𝐴 dom 𝑔 = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
2217, 18, 213eqtri 2195 . . 3 dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
23 ordeq 4357 . . 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 145 1 Ord dom recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  {cab 2156  wral 2448  wrex 2449  wss 3121   cuni 3796   ciun 3873  Ord word 4347  Oncon0 4348  dom cdm 4611  cres 4613   Fn wfn 5193  cfv 5198  recscrecs 6283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-tr 4088  df-iord 4351  df-on 4353  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-recs 6284
This theorem is referenced by:  tfrlemi14d  6312  tfri1dALT  6330
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