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Theorem rdgprc 32925
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 6666 . . . . . . 7 (𝑧 = ∅ → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘∅))
2 fveq2 6666 . . . . . . 7 (𝑧 = ∅ → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘∅))
31, 2eqeq12d 2841 . . . . . 6 (𝑧 = ∅ → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅)))
43imbi2d 342 . . . . 5 (𝑧 = ∅ → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))))
5 fveq2 6666 . . . . . . 7 (𝑧 = 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑦))
6 fveq2 6666 . . . . . . 7 (𝑧 = 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑦))
75, 6eqeq12d 2841 . . . . . 6 (𝑧 = 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
87imbi2d 342 . . . . 5 (𝑧 = 𝑦 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦))))
9 fveq2 6666 . . . . . . 7 (𝑧 = suc 𝑦 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘suc 𝑦))
10 fveq2 6666 . . . . . . 7 (𝑧 = suc 𝑦 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘suc 𝑦))
119, 10eqeq12d 2841 . . . . . 6 (𝑧 = suc 𝑦 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦)))
1211imbi2d 342 . . . . 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 6666 . . . . . . 7 (𝑧 = 𝑥 → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼)‘𝑥))
14 fveq2 6666 . . . . . . 7 (𝑧 = 𝑥 → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅)‘𝑥))
1513, 14eqeq12d 2841 . . . . . 6 (𝑧 = 𝑥 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥)))
1615imbi2d 342 . . . . 5 (𝑧 = 𝑥 → ((¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)) ↔ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))))
17 rdgprc0 32924 . . . . . 6 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)
18 0ex 5207 . . . . . . 7 ∅ ∈ V
1918rdg0 8051 . . . . . 6 (rec(𝐹, ∅)‘∅) = ∅
2017, 19syl6eqr 2878 . . . . 5 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = (rec(𝐹, ∅)‘∅))
21 fveq2 6666 . . . . . . 7 ((rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦) → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
22 rdgsuc 8054 . . . . . . . 8 (𝑦 ∈ On → (rec(𝐹, 𝐼)‘suc 𝑦) = (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
23 rdgsuc 8054 . . . . . . . 8 (𝑦 ∈ On → (rec(𝐹, ∅)‘suc 𝑦) = (𝐹‘(rec(𝐹, ∅)‘𝑦)))
2422, 23eqeq12d 2841 . . . . . . 7 (𝑦 ∈ On → ((rec(𝐹, 𝐼)‘suc 𝑦) = (rec(𝐹, ∅)‘suc 𝑦) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, ∅)‘𝑦))))
2521, 24syl5ibr 247 . . . . . 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 3179 . . . . . 6 (∀𝑦𝑧𝐼 ∈ V → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)) ↔ (¬ 𝐼 ∈ V → ∀𝑦𝑧 (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, ∅)‘𝑦)))
28 limord 6247 . . . . . . . . 9 (Lim 𝑧 → Ord 𝑧)
29 ordsson 7495 . . . . . . . . 9 (Ord 𝑧𝑧 ⊆ On)
30 rdgfnon 8048 . . . . . . . . . 10 rec(𝐹, 𝐼) Fn On
31 rdgfnon 8048 . . . . . . . . . 10 rec(𝐹, ∅) Fn On
32 fvreseq 6805 . . . . . . . . . 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 698 . . . . . . . . 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 5804 . . . . . . . . . . 11 ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → ran (rec(𝐹, 𝐼) ↾ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧))
36 df-ima 5566 . . . . . . . . . . 11 (rec(𝐹, 𝐼) “ 𝑧) = ran (rec(𝐹, 𝐼) ↾ 𝑧)
37 df-ima 5566 . . . . . . . . . . 11 (rec(𝐹, ∅) “ 𝑧) = ran (rec(𝐹, ∅) ↾ 𝑧)
3835, 36, 373eqtr4g 2885 . . . . . . . . . 10 ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
3938unieqd 4846 . . . . . . . . 9 ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧))
40 vex 3502 . . . . . . . . . 10 𝑧 ∈ V
41 rdglim 8056 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, 𝐼) “ 𝑧))
42 rdglim 8056 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → (rec(𝐹, ∅)‘𝑧) = (rec(𝐹, ∅) “ 𝑧))
4341, 42eqeq12d 2841 . . . . . . . . . 10 ((𝑧 ∈ V ∧ Lim 𝑧) → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧)))
4440, 43mpan 686 . . . . . . . . 9 (Lim 𝑧 → ((rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧) ↔ (rec(𝐹, 𝐼) “ 𝑧) = (rec(𝐹, ∅) “ 𝑧)))
4539, 44syl5ibr 247 . . . . . . . 8 (Lim 𝑧 → ((rec(𝐹, 𝐼) ↾ 𝑧) = (rec(𝐹, ∅) ↾ 𝑧) → (rec(𝐹, 𝐼)‘𝑧) = (rec(𝐹, ∅)‘𝑧)))
4634, 45sylbird 261 . . . . . . 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 243 . . . . 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 7565 . . . 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 3185 . 2 𝐼 ∈ V → ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
52 eqfnfv 6797 . . 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 688 . 2 (rec(𝐹, 𝐼) = rec(𝐹, ∅) ↔ ∀𝑥 ∈ On (rec(𝐹, 𝐼)‘𝑥) = (rec(𝐹, ∅)‘𝑥))
5451, 53sylibr 235 1 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499  wss 3939  c0 4294   cuni 4836  ran crn 5554  cres 5555  cima 5556  Ord word 6187  Oncon0 6188  Lim wlim 6189  suc csuc 6190   Fn wfn 6346  cfv 6351  reccrdg 8039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040
This theorem is referenced by:  dfrdg3  32927
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