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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
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