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Theorem tfrlem9 5876
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 4474 . . 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 5861 . . . . . 6 recs(𝐹) = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
43eleq2i 2101 . . . . 5 (⟨B, z recs(𝐹) ↔ ⟨B, z {fx On (f Fn x y x (fy) = (𝐹‘(fy)))})
5 eluniab 3583 . . . . 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 4945 . . . . . . . . . . . . . 14 ((f Fn x B, z f) → B x)
8 rspe 2364 . . . . . . . . . . . . . . . 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 2145 . . . . . . . . . . . . . . . . 17 (f Ax On (f Fn x y x (fy) = (𝐹‘(fy))))
11 elssuni 3599 . . . . . . . . . . . . . . . . . 18 (f Af A)
129recsfval 5872 . . . . . . . . . . . . . . . . . 18 recs(𝐹) = A
1311, 12syl6sseqr 2986 . . . . . . . . . . . . . . . . 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 5121 . . . . . . . . . . . . . . . . . . . 20 (y = B → (fy) = (fB))
17 reseq2 4550 . . . . . . . . . . . . . . . . . . . . 21 (y = B → (fy) = (fB))
1817fveq2d 5125 . . . . . . . . . . . . . . . . . . . 20 (y = B → (𝐹‘(fy)) = (𝐹‘(fB)))
1916, 18eqeq12d 2051 . . . . . . . . . . . . . . . . . . 19 (y = B → ((fy) = (𝐹‘(fy)) ↔ (fB) = (𝐹‘(fB))))
2019rspcv 2646 . . . . . . . . . . . . . . . . . 18 (B x → (y x (fy) = (𝐹‘(fy)) → (fB) = (𝐹‘(fB))))
21 fndm 4941 . . . . . . . . . . . . . . . . . . . . 21 (f Fn x → dom f = x)
2221eleq2d 2104 . . . . . . . . . . . . . . . . . . . 20 (f Fn x → (B dom fB x))
239tfrlem7 5874 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Fun recs(𝐹)
24 funssfv 5142 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) f ⊆ recs(𝐹) B dom f) → (recs(𝐹)‘B) = (fB))
2523, 24mp3an1 1218 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((f ⊆ recs(𝐹) B dom f) → (recs(𝐹)‘B) = (fB))
2625adantrl 447 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((f ⊆ recs(𝐹) ((f Fn x x On) B dom f)) → (recs(𝐹)‘B) = (fB))
2721eleq1d 2103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (f Fn x → (dom f On ↔ x On))
28 onelss 4090 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun recs(𝐹) f ⊆ recs(𝐹) B ⊆ dom f) → (recs(𝐹) ↾ B) = (fB))
3231fveq2d 5125 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun recs(𝐹) f ⊆ recs(𝐹) B ⊆ dom f) → (𝐹‘(recs(𝐹) ↾ B)) = (𝐹‘(fB)))
3323, 32mp3an1 1218 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2051 . . . . . . . . . . . . . . . . . . . . . . . . 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 2426 . . . . . 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 1486 . . . 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 1486 . 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 884   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  wss 2911  cop 3370   cuni 3571  Oncon0 4066  dom cdm 4288  cres 4290  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:  tfr2a  5877  tfrlemiubacc  5885
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