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

Theorem tfrlemiubacc 5885
 Description: The union of B satisfies the recursion rule (lemma for tfrlemi1 5887). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
tfrlemisucfn.2 (φx(Fun 𝐹 (𝐹x) V))
tfrlemi1.3 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
tfrlemi1.4 (φx On)
tfrlemi1.5 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
Assertion
Ref Expression
tfrlemiubacc (φu x ( Bu) = (𝐹‘( Bu)))
Distinct variable groups:   f,g,,u,w,x,y,z,A   f,𝐹,g,,u,w,x,y,z   φ,w,y   u,B,w,f,g,,z   φ,g,,z
Allowed substitution hints:   φ(x,u,f)   B(x,y)

Proof of Theorem tfrlemiubacc
StepHypRef Expression
1 tfrlemisucfn.1 . . . . . . . . 9 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
2 tfrlemisucfn.2 . . . . . . . . 9 (φx(Fun 𝐹 (𝐹x) V))
3 tfrlemi1.3 . . . . . . . . 9 B = {z x g(g Fn z g A = (g ∪ {⟨z, (𝐹g)⟩}))}
4 tfrlemi1.4 . . . . . . . . 9 (φx On)
5 tfrlemi1.5 . . . . . . . . 9 (φz x g(g Fn z w z (gw) = (𝐹‘(gw))))
61, 2, 3, 4, 5tfrlemibfn 5883 . . . . . . . 8 (φ B Fn x)
7 fndm 4941 . . . . . . . 8 ( B Fn x → dom B = x)
86, 7syl 14 . . . . . . 7 (φ → dom B = x)
91, 2, 3, 4, 5tfrlemibacc 5881 . . . . . . . . . 10 (φBA)
109unissd 3595 . . . . . . . . 9 (φ B A)
111recsfval 5872 . . . . . . . . 9 recs(𝐹) = A
1210, 11syl6sseqr 2986 . . . . . . . 8 (φ B ⊆ recs(𝐹))
13 dmss 4477 . . . . . . . 8 ( B ⊆ recs(𝐹) → dom B ⊆ dom recs(𝐹))
1412, 13syl 14 . . . . . . 7 (φ → dom B ⊆ dom recs(𝐹))
158, 14eqsstr3d 2974 . . . . . 6 (φx ⊆ dom recs(𝐹))
1615sselda 2939 . . . . 5 ((φ w x) → w dom recs(𝐹))
171tfrlem9 5876 . . . . 5 (w dom recs(𝐹) → (recs(𝐹)‘w) = (𝐹‘(recs(𝐹) ↾ w)))
1816, 17syl 14 . . . 4 ((φ w x) → (recs(𝐹)‘w) = (𝐹‘(recs(𝐹) ↾ w)))
191tfrlem7 5874 . . . . . 6 Fun recs(𝐹)
2019a1i 9 . . . . 5 ((φ w x) → Fun recs(𝐹))
2112adantr 261 . . . . 5 ((φ w x) → B ⊆ recs(𝐹))
228eleq2d 2104 . . . . . 6 (φ → (w dom Bw x))
2322biimpar 281 . . . . 5 ((φ w x) → w dom B)
24 funssfv 5142 . . . . 5 ((Fun recs(𝐹) B ⊆ recs(𝐹) w dom B) → (recs(𝐹)‘w) = ( Bw))
2520, 21, 23, 24syl3anc 1134 . . . 4 ((φ w x) → (recs(𝐹)‘w) = ( Bw))
26 eloni 4078 . . . . . . . . 9 (x On → Ord x)
274, 26syl 14 . . . . . . . 8 (φ → Ord x)
28 ordelss 4082 . . . . . . . 8 ((Ord x w x) → wx)
2927, 28sylan 267 . . . . . . 7 ((φ w x) → wx)
308adantr 261 . . . . . . 7 ((φ w x) → dom B = x)
3129, 30sseqtr4d 2976 . . . . . 6 ((φ w x) → w ⊆ dom B)
32 fun2ssres 4886 . . . . . 6 ((Fun recs(𝐹) B ⊆ recs(𝐹) w ⊆ dom B) → (recs(𝐹) ↾ w) = ( Bw))
3320, 21, 31, 32syl3anc 1134 . . . . 5 ((φ w x) → (recs(𝐹) ↾ w) = ( Bw))
3433fveq2d 5125 . . . 4 ((φ w x) → (𝐹‘(recs(𝐹) ↾ w)) = (𝐹‘( Bw)))
3518, 25, 343eqtr3d 2077 . . 3 ((φ w x) → ( Bw) = (𝐹‘( Bw)))
3635ralrimiva 2386 . 2 (φw x ( Bw) = (𝐹‘( Bw)))
37 fveq2 5121 . . . 4 (u = w → ( Bu) = ( Bw))
38 reseq2 4550 . . . . 5 (u = w → ( Bu) = ( Bw))
3938fveq2d 5125 . . . 4 (u = w → (𝐹‘( Bu)) = (𝐹‘( Bw)))
4037, 39eqeq12d 2051 . . 3 (u = w → (( Bu) = (𝐹‘( Bu)) ↔ ( Bw) = (𝐹‘( Bw))))
4140cbvralv 2527 . 2 (u x ( Bu) = (𝐹‘( Bu)) ↔ w x ( Bw) = (𝐹‘( Bw)))
4236, 41sylibr 137 1 (φu x ( Bu) = (𝐹‘( Bu)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  {csn 3367  ⟨cop 3370  ∪ cuni 3571  Ord word 4065  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-in1 544  ax-in2 545  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  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  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-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  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-suc 4074  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-f 4849  df-fv 4853  df-recs 5861 This theorem is referenced by:  tfrlemiex  5886
 Copyright terms: Public domain W3C validator