Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem9 Structured version   GIF version

Theorem tfrlem9 5857
 Description: Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
Hypothesis
Ref Expression
tfrlem.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Assertion
Ref Expression
tfrlem9 (B dom recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
Distinct variable groups:   x,f,y,B   f,𝐹,x,y
Allowed substitution hints:   A(x,y,f)

Proof of Theorem tfrlem9
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eldm2g 4458 . . 3 (B dom recs(𝐹) → (B dom recs(𝐹) ↔ zB, z recs(𝐹)))
21ibi 165 . 2 (B dom recs(𝐹) → zB, z recs(𝐹))
3 df-recs 5842 . . . . . 6 recs(𝐹) = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
43eleq2i 2086 . . . . 5 (⟨B, z recs(𝐹) ↔ ⟨B, z {fx On (f Fn x y x (fy) = (𝐹‘(fy)))})
5 eluniab 3566 . . . . 5 (⟨B, z {fx On (f Fn x y x (fy) = (𝐹‘(fy)))} ↔ f(⟨B, z f x On (f Fn x y x (fy) = (𝐹‘(fy)))))
64, 5bitri 173 . . . 4 (⟨B, z recs(𝐹) ↔ f(⟨B, z f x On (f Fn x y x (fy) = (𝐹‘(fy)))))
7 fnop 4928 . . . . . . . . . . . . . 14 ((f Fn x B, z f) → B x)
8 rspe 2348 . . . . . . . . . . . . . . . 16 ((x On (f Fn x y x (fy) = (𝐹‘(fy)))) → x On (f Fn x y x (fy) = (𝐹‘(fy))))
9 tfrlem.1 . . . . . . . . . . . . . . . . . 18 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
109abeq2i 2130 . . . . . . . . . . . . . . . . 17 (f Ax On (f Fn x y x (fy) = (𝐹‘(fy))))
11 elssuni 3582 . . . . . . . . . . . . . . . . . 18 (f Af A)
129recsfval 5853 . . . . . . . . . . . . . . . . . 18 recs(𝐹) = A
1311, 12syl6sseqr 2969 . . . . . . . . . . . . . . . . 17 (f Af ⊆ recs(𝐹))
1410, 13sylbir 125 . . . . . . . . . . . . . . . 16 (x On (f Fn x y x (fy) = (𝐹‘(fy))) → f ⊆ recs(𝐹))
158, 14syl 14 . . . . . . . . . . . . . . 15 ((x On (f Fn x y x (fy) = (𝐹‘(fy)))) → f ⊆ recs(𝐹))
16 fveq2 5103 . . . . . . . . . . . . . . . . . . . 20 (y = B → (fy) = (fB))
17 reseq2 4534 . . . . . . . . . . . . . . . . . . . . 21 (y = B → (fy) = (fB))
1817fveq2d 5107 . . . . . . . . . . . . . . . . . . . 20 (y = B → (𝐹‘(fy)) = (𝐹‘(fB)))
1916, 18eqeq12d 2036 . . . . . . . . . . . . . . . . . . 19 (y = B → ((fy) = (𝐹‘(fy)) ↔ (fB) = (𝐹‘(fB))))
2019rspcv 2629 . . . . . . . . . . . . . . . . . 18 (B x → (y x (fy) = (𝐹‘(fy)) → (fB) = (𝐹‘(fB))))
21 fndm 4924 . . . . . . . . . . . . . . . . . . . . 21 (f Fn x → dom f = x)
2221eleq2d 2089 . . . . . . . . . . . . . . . . . . . 20 (f Fn x → (B dom fB x))
239tfrlem7 5855 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Fun recs(𝐹)
24 funssfv 5124 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) f ⊆ recs(𝐹) B dom f) → (recs(𝐹)‘B) = (fB))
2523, 24mp3an1 1204 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((f ⊆ recs(𝐹) B dom f) → (recs(𝐹)‘B) = (fB))
2625adantrl 450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((f ⊆ recs(𝐹) ((f Fn x x On) B dom f)) → (recs(𝐹)‘B) = (fB))
2721eleq1d 2088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (f Fn x → (dom f On ↔ x On))
28 onelss 4073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom f On → (B dom fB ⊆ dom f))
2927, 28syl6bir 153 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (f Fn x → (x On → (B dom fB ⊆ dom f)))
3029imp31 243 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((f Fn x x On) B dom f) → B ⊆ dom f)
31 fun2ssres 4869 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun recs(𝐹) f ⊆ recs(𝐹) B ⊆ dom f) → (recs(𝐹) ↾ B) = (fB))
3231fveq2d 5107 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) f ⊆ recs(𝐹) B ⊆ dom f) → (𝐹‘(recs(𝐹) ↾ B)) = (𝐹‘(fB)))
3323, 32mp3an1 1204 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((f ⊆ recs(𝐹) B ⊆ dom f) → (𝐹‘(recs(𝐹) ↾ B)) = (𝐹‘(fB)))
3430, 33sylan2 270 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((f ⊆ recs(𝐹) ((f Fn x x On) B dom f)) → (𝐹‘(recs(𝐹) ↾ B)) = (𝐹‘(fB)))
3526, 34eqeq12d 2036 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((f ⊆ recs(𝐹) ((f Fn x x On) B dom f)) → ((recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)) ↔ (fB) = (𝐹‘(fB))))
3635exbiri 364 . . . . . . . . . . . . . . . . . . . . . . . 24 (f ⊆ recs(𝐹) → (((f Fn x x On) B dom f) → ((fB) = (𝐹‘(fB)) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))
3736com3l 75 . . . . . . . . . . . . . . . . . . . . . . 23 (((f Fn x x On) B dom f) → ((fB) = (𝐹‘(fB)) → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))
3837exp31 346 . . . . . . . . . . . . . . . . . . . . . 22 (f Fn x → (x On → (B dom f → ((fB) = (𝐹‘(fB)) → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
3938com34 77 . . . . . . . . . . . . . . . . . . . . 21 (f Fn x → (x On → ((fB) = (𝐹‘(fB)) → (B dom f → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
4039com24 81 . . . . . . . . . . . . . . . . . . . 20 (f Fn x → (B dom f → ((fB) = (𝐹‘(fB)) → (x On → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
4122, 40sylbird 159 . . . . . . . . . . . . . . . . . . 19 (f Fn x → (B x → ((fB) = (𝐹‘(fB)) → (x On → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
4241com3l 75 . . . . . . . . . . . . . . . . . 18 (B x → ((fB) = (𝐹‘(fB)) → (f Fn x → (x On → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
4320, 42syld 40 . . . . . . . . . . . . . . . . 17 (B x → (y x (fy) = (𝐹‘(fy)) → (f Fn x → (x On → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
4443com24 81 . . . . . . . . . . . . . . . 16 (B x → (x On → (f Fn x → (y x (fy) = (𝐹‘(fy)) → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
4544imp4d 334 . . . . . . . . . . . . . . 15 (B x → ((x On (f Fn x y x (fy) = (𝐹‘(fy)))) → (f ⊆ recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))
4615, 45mpdi 38 . . . . . . . . . . . . . 14 (B x → ((x On (f Fn x y x (fy) = (𝐹‘(fy)))) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B))))
477, 46syl 14 . . . . . . . . . . . . 13 ((f Fn x B, z f) → ((x On (f Fn x y x (fy) = (𝐹‘(fy)))) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B))))
4847exp4d 351 . . . . . . . . . . . 12 ((f Fn x B, z f) → (x On → (f Fn x → (y x (fy) = (𝐹‘(fy)) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B))))))
4948ex 108 . . . . . . . . . . 11 (f Fn x → (⟨B, z f → (x On → (f Fn x → (y x (fy) = (𝐹‘(fy)) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
5049com4r 80 . . . . . . . . . 10 (f Fn x → (f Fn x → (⟨B, z f → (x On → (y x (fy) = (𝐹‘(fy)) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))))
5150pm2.43i 43 . . . . . . . . 9 (f Fn x → (⟨B, z f → (x On → (y x (fy) = (𝐹‘(fy)) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B))))))
5251com3l 75 . . . . . . . 8 (⟨B, z f → (x On → (f Fn x → (y x (fy) = (𝐹‘(fy)) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B))))))
5352imp4a 331 . . . . . . 7 (⟨B, z f → (x On → ((f Fn x y x (fy) = (𝐹‘(fy))) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))))
5453rexlimdv 2410 . . . . . 6 (⟨B, z f → (x On (f Fn x y x (fy) = (𝐹‘(fy))) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B))))
5554imp 115 . . . . 5 ((⟨B, z f x On (f Fn x y x (fy) = (𝐹‘(fy)))) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
5655exlimiv 1471 . . . 4 (f(⟨B, z f x On (f Fn x y x (fy) = (𝐹‘(fy)))) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
576, 56sylbi 114 . . 3 (⟨B, z recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
5857exlimiv 1471 . 2 (zB, z recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
592, 58syl 14 1 (B dom recs(𝐹) → (recs(𝐹)‘B) = (𝐹‘(recs(𝐹) ↾ B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 873   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008  ∀wral 2284  ∃wrex 2285   ⊆ wss 2894  ⟨cop 3353  ∪ cuni 3554  Oncon0 4049  dom cdm 4272   ↾ cres 4274  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829  recscrecs 5841 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-recs 5842 This theorem is referenced by:  tfr2a  5858  tfrlemiubacc  5865
 Copyright terms: Public domain W3C validator