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Theorem tfrlem8 7649
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 7643 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤)))}
32abeq2i 2873 . . . . . . 7 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
4 fndm 6151 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
54adantr 472 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 = 𝑧)
65eleq1d 2824 . . . . . . . . 9 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On))
76biimprcd 240 . . . . . . . 8 (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On))
87rexlimiv 3165 . . . . . . 7 (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))) → dom 𝑔 ∈ On)
93, 8sylbi 207 . . . . . 6 (𝑔𝐴 → dom 𝑔 ∈ On)
10 eleq1a 2834 . . . . . 6 (dom 𝑔 ∈ On → (𝑧 = dom 𝑔𝑧 ∈ On))
119, 10syl 17 . . . . 5 (𝑔𝐴 → (𝑧 = dom 𝑔𝑧 ∈ On))
1211rexlimiv 3165 . . . 4 (∃𝑔𝐴 𝑧 = dom 𝑔𝑧 ∈ On)
1312abssi 3818 . . 3 {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔} ⊆ On
14 ssorduni 7150 . . 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 7646 . . . . 5 recs(𝐹) = 𝐴
1716dmeqi 5480 . . . 4 dom recs(𝐹) = dom 𝐴
18 dmuni 5489 . . . 4 dom 𝐴 = 𝑔𝐴 dom 𝑔
19 vex 3343 . . . . . 6 𝑔 ∈ V
2019dmex 7264 . . . . 5 dom 𝑔 ∈ V
2120dfiun2 4706 . . . 4 𝑔𝐴 dom 𝑔 = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
2217, 18, 213eqtri 2786 . . 3 dom recs(𝐹) = {𝑧 ∣ ∃𝑔𝐴 𝑧 = dom 𝑔}
23 ordeq 5891 . . 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 221 1 Ord dom recs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  wss 3715   cuni 4588   ciun 4672  dom cdm 5266  cres 5268  Ord word 5883  Oncon0 5884   Fn wfn 6044  cfv 6049  recscrecs 7636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-wrecs 7576  df-recs 7637
This theorem is referenced by:  tfrlem10  7652  tfrlem12  7654  tfrlem13  7655  tfrlem14  7656  tfrlem15  7657  tfrlem16  7658
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