ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem7 Structured version   GIF version

Theorem tfrlem7 5874
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Assertion
Ref Expression
tfrlem7 Fun recs(𝐹)
Distinct variable group:   x,f,y,𝐹
Allowed substitution hints:   A(x,y,f)

Proof of Theorem tfrlem7
Dummy variables g u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
21tfrlem6 5873 . 2 Rel recs(𝐹)
31recsfval 5872 . . . . . . . . 9 recs(𝐹) = A
43eleq2i 2101 . . . . . . . 8 (⟨x, u recs(𝐹) ↔ ⟨x, u A)
5 eluni 3574 . . . . . . . 8 (⟨x, u Ag(⟨x, u g g A))
64, 5bitri 173 . . . . . . 7 (⟨x, u recs(𝐹) ↔ g(⟨x, u g g A))
73eleq2i 2101 . . . . . . . 8 (⟨x, v recs(𝐹) ↔ ⟨x, v A)
8 eluni 3574 . . . . . . . 8 (⟨x, v A(⟨x, v A))
97, 8bitri 173 . . . . . . 7 (⟨x, v recs(𝐹) ↔ (⟨x, v A))
106, 9anbi12i 433 . . . . . 6 ((⟨x, u recs(𝐹) x, v recs(𝐹)) ↔ (g(⟨x, u g g A) (⟨x, v A)))
11 eeanv 1804 . . . . . 6 (g((⟨x, u g g A) (⟨x, v A)) ↔ (g(⟨x, u g g A) (⟨x, v A)))
1210, 11bitr4i 176 . . . . 5 ((⟨x, u recs(𝐹) x, v recs(𝐹)) ↔ g((⟨x, u g g A) (⟨x, v A)))
13 df-br 3756 . . . . . . . . 9 (xgu ↔ ⟨x, u g)
14 df-br 3756 . . . . . . . . 9 (xv ↔ ⟨x, v )
1513, 14anbi12i 433 . . . . . . . 8 ((xgu xv) ↔ (⟨x, u g x, v ))
161tfrlem5 5871 . . . . . . . . 9 ((g A A) → ((xgu xv) → u = v))
1716impcom 116 . . . . . . . 8 (((xgu xv) (g A A)) → u = v)
1815, 17sylanbr 269 . . . . . . 7 (((⟨x, u g x, v ) (g A A)) → u = v)
1918an4s 522 . . . . . 6 (((⟨x, u g g A) (⟨x, v A)) → u = v)
2019exlimivv 1773 . . . . 5 (g((⟨x, u g g A) (⟨x, v A)) → u = v)
2112, 20sylbi 114 . . . 4 ((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)
2221ax-gen 1335 . . 3 v((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)
2322gen2 1336 . 2 xuv((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)
24 dffun4 4856 . 2 (Fun recs(𝐹) ↔ (Rel recs(𝐹) xuv((⟨x, u recs(𝐹) x, v recs(𝐹)) → u = v)))
252, 23, 24mpbir2an 848 1 Fun recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  cop 3370   cuni 3571   class class class wbr 3755  Oncon0 4066  cres 4290  Rel wrel 4293  Fun wfun 4839   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-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-setind 4220
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-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfrlem9  5876  tfrlemibfn  5883  tfrlemiubacc  5885  tfri1d  5890  rdgfun  5900
  Copyright terms: Public domain W3C validator