Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rdgprc Structured version   Visualization version   GIF version

Theorem rdgprc 31824
Description: The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rdgprc 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Proof of Theorem rdgprc
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . . . 7 (𝑧 = ∅ → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘∅))
2 fveq2 6229 . . . . . . 7 (𝑧 = ∅ → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘∅))
31, 2eqeq12d 2666 . . . . . 6 (𝑧 = ∅ → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅)))
43imbi2d 329 . . . . 5 (𝑧 = ∅ → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))))
5 fveq2 6229 . . . . . . 7 (𝑧 = 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑦))
6 fveq2 6229 . . . . . . 7 (𝑧 = 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑦))
75, 6eqeq12d 2666 . . . . . 6 (𝑧 = 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
87imbi2d 329 . . . . 5 (𝑧 = 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦))))
9 fveq2 6229 . . . . . . 7 (𝑧 = suc 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘suc 𝑦))
10 fveq2 6229 . . . . . . 7 (𝑧 = suc 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘suc 𝑦))
119, 10eqeq12d 2666 . . . . . 6 (𝑧 = suc 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
1211imbi2d 329 . . . . 5 (𝑧 = suc 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦))))
13 fveq2 6229 . . . . . . 7 (𝑧 = 𝑥 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑥))
14 fveq2 6229 . . . . . . 7 (𝑧 = 𝑥 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑥))
1513, 14eqeq12d 2666 . . . . . 6 (𝑧 = 𝑥 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
1615imbi2d 329 . . . . 5 (𝑧 = 𝑥 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))))
17 rdgprc0 31823 . . . . . 6 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
18 0ex 4823 . . . . . . 7 ∅ ∈ V
1918rdg0 7562 . . . . . 6 (rec(𝐹, ∅)‘∅) = ∅
2017, 19syl6eqr 2703 . . . . 5 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))
21 fveq2 6229 . . . . . . 7 ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
22 rdgsuc 7565 . . . . . . . 8 (𝑦 ∈ On → (rec(𝐹, 𝐼)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
23 rdgsuc 7565 . . . . . . . 8 (𝑦 ∈ On → (rec(𝐹, ∅)‘suc 𝑦) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
2422, 23eqeq12d 2666 . . . . . . 7 (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦))))
2521, 24syl5ibr 236 . . . . . 6 (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
2625imim2d 57 . . . . 5 (𝑦 ∈ On → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦))))
27 r19.21v 2989 . . . . . 6 (∀𝑦𝑧𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) ↔ (¬ 𝐼 ∈ V → ∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
28 limord 5822 . . . . . . . . 9 (Lim 𝑧 → Ord 𝑧)
29 ordsson 7031 . . . . . . . . 9 (Ord 𝑧𝑧 ⊆ On)
30 rdgfnon 7559 . . . . . . . . . 10 rec(𝐹, 𝐼) Fn On
31 rdgfnon 7559 . . . . . . . . . 10 rec(𝐹, ∅) Fn On
32 fvreseq 6359 . . . . . . . . . 10 (((rec(𝐹, 𝐼) Fn On ∧ rec(𝐹, ∅) Fn On) ∧ 𝑧 ⊆ On) → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
3330, 31, 32mpanl12 718 . . . . . . . . 9 (𝑧 ⊆ On → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
3428, 29, 333syl 18 . . . . . . . 8 (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) ↔ ∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
35 rneq 5383 . . . . . . . . . . 11 ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ran (rec(𝐹, 𝐼) ↾ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧))
36 df-ima 5156 . . . . . . . . . . 11 (rec(𝐹, 𝐼) “ 𝑧) = ran (rec(𝐹, 𝐼) ↾ 𝑧)
37 df-ima 5156 . . . . . . . . . . 11 (rec(𝐹, ∅) “ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧)
3835, 36, 373eqtr4g 2710 . . . . . . . . . 10 ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
3938unieqd 4478 . . . . . . . . 9 ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
40 vex 3234 . . . . . . . . . 10 𝑧 ∈ V
41 rdglim 7567 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼) “ 𝑧))
42 rdglim 7567 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅) “ 𝑧))
4341, 42eqeq12d 2666 . . . . . . . . . 10 ((𝑧 ∈ V ∧ Lim 𝑧) → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧)))
4440, 43mpan 706 . . . . . . . . 9 (Lim 𝑧 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧)))
4539, 44syl5ibr 236 . . . . . . . 8 (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
4634, 45sylbird 250 . . . . . . 7 (Lim 𝑧 → (∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
4746imim2d 57 . . . . . 6 (Lim 𝑧 → ((¬ 𝐼 ∈ V → ∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧))))
4827, 47syl5bi 232 . . . . 5 (Lim 𝑧 → (∀𝑦𝑧𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧))))
494, 8, 12, 16, 20, 26, 48tfinds 7101 . . . 4 (𝑥 ∈ On → (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
5049com12 32 . . 3 𝐼 ∈ V → (𝑥 ∈ On → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
5150ralrimiv 2994 . 2 𝐼 ∈ V → ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
52 eqfnfv 6351 . . 3 ((rec(𝐹, 𝐼) Fn On ∧ rec(𝐹, ∅) Fn On) → (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
5330, 31, 52mp2an 708 . 2 (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
5451, 53sylibr 224 1 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948   cuni 4468  ran crn 5144  cres 5145  cima 5146  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  cfv 5926  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  dfrdg3  31826
  Copyright terms: Public domain W3C validator