ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem8 Structured version   GIF version

Theorem tfrlem8 5875
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 = {fx On (f Fn x y x (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 = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
21tfrlem3 5867 . . . . . . . 8 A = {gz On (g Fn z w z (gw) = (𝐹‘(gw)))}
32abeq2i 2145 . . . . . . 7 (g Az On (g Fn z w z (gw) = (𝐹‘(gw))))
4 fndm 4941 . . . . . . . . . . 11 (g Fn z → dom g = z)
54adantr 261 . . . . . . . . . 10 ((g Fn z w z (gw) = (𝐹‘(gw))) → dom g = z)
65eleq1d 2103 . . . . . . . . 9 ((g Fn z w z (gw) = (𝐹‘(gw))) → (dom g On ↔ z On))
76biimprcd 149 . . . . . . . 8 (z On → ((g Fn z w z (gw) = (𝐹‘(gw))) → dom g On))
87rexlimiv 2421 . . . . . . 7 (z On (g Fn z w z (gw) = (𝐹‘(gw))) → dom g On)
93, 8sylbi 114 . . . . . 6 (g A → dom g On)
10 eleq1a 2106 . . . . . 6 (dom g On → (z = dom gz On))
119, 10syl 14 . . . . 5 (g A → (z = dom gz On))
1211rexlimiv 2421 . . . 4 (g A z = dom gz On)
1312abssi 3009 . . 3 {zg A z = dom g} ⊆ On
14 ssorduni 4179 . . 3 ({zg A z = dom g} ⊆ On → Ord {zg A z = dom g})
1513, 14ax-mp 7 . 2 Ord {zg A z = dom g}
161recsfval 5872 . . . . 5 recs(𝐹) = A
1716dmeqi 4479 . . . 4 dom recs(𝐹) = dom A
18 dmuni 4488 . . . 4 dom A = g A dom g
19 vex 2554 . . . . . 6 g V
2019dmex 4541 . . . . 5 dom g V
2120dfiun2 3682 . . . 4 g A dom g = {zg A z = dom g}
2217, 18, 213eqtri 2061 . . 3 dom recs(𝐹) = {zg A z = dom g}
23 ordeq 4075 . . 3 (dom recs(𝐹) = {zg A z = dom g} → (Ord dom recs(𝐹) ↔ Ord {zg A z = dom g}))
2422, 23ax-mp 7 . 2 (Ord dom recs(𝐹) ↔ Ord {zg A z = dom g})
2515, 24mpbir 134 1 Ord dom recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  wss 2911   cuni 3571   ciun 3648  Ord word 4065  Oncon0 4066  dom cdm 4288  cres 4290   Fn wfn 4840  cfv 4845  recscrecs 5860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfrlemi14d  5888
  Copyright terms: Public domain W3C validator