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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
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