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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
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