MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem12 Structured version   Visualization version   GIF version

Theorem tfrlem12 8419
Description: Lemma for transfinite recursion. Show 𝐶 is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlem.3 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
Assertion
Ref Expression
tfrlem12 (recs(𝐹) ∈ V → 𝐶𝐴)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐶   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem12
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . 6 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem8 8414 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V → Ord dom recs(𝐹))
4 dmexg 7914 . . . 4 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
5 elon2 6387 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 581 . . 3 (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
7 onsuc 7820 . . . 4 (dom recs(𝐹) ∈ On → suc dom recs(𝐹) ∈ On)
8 tfrlem.3 . . . . 5 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
91, 8tfrlem10 8417 . . . 4 (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))
101, 8tfrlem11 8418 . . . . . 6 (dom recs(𝐹) ∈ On → (𝑧 ∈ suc dom recs(𝐹) → (𝐶𝑧) = (𝐹‘(𝐶𝑧))))
1110ralrimiv 3135 . . . . 5 (dom recs(𝐹) ∈ On → ∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)))
12 fveq2 6901 . . . . . . 7 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
13 reseq2 5984 . . . . . . . 8 (𝑧 = 𝑦 → (𝐶𝑧) = (𝐶𝑦))
1413fveq2d 6905 . . . . . . 7 (𝑧 = 𝑦 → (𝐹‘(𝐶𝑧)) = (𝐹‘(𝐶𝑦)))
1512, 14eqeq12d 2742 . . . . . 6 (𝑧 = 𝑦 → ((𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
1615cbvralvw 3225 . . . . 5 (∀𝑧 ∈ suc dom recs(𝐹)(𝐶𝑧) = (𝐹‘(𝐶𝑧)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
1711, 16sylib 217 . . . 4 (dom recs(𝐹) ∈ On → ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))
18 fneq2 6652 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (𝐶 Fn 𝑥𝐶 Fn suc dom recs(𝐹)))
19 raleq 3312 . . . . . 6 (𝑥 = suc dom recs(𝐹) → (∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)) ↔ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦))))
2018, 19anbi12d 630 . . . . 5 (𝑥 = suc dom recs(𝐹) → ((𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))) ↔ (𝐶 Fn suc dom recs(𝐹) ∧ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
2120rspcev 3608 . . . 4 ((suc dom recs(𝐹) ∈ On ∧ (𝐶 Fn suc dom recs(𝐹) ∧ ∀𝑦 ∈ suc dom recs(𝐹)(𝐶𝑦) = (𝐹‘(𝐶𝑦)))) → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
227, 9, 17, 21syl12anc 835 . . 3 (dom recs(𝐹) ∈ On → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
236, 22syl 17 . 2 (recs(𝐹) ∈ V → ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
24 snex 5437 . . . . 5 {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V
25 unexg 7757 . . . . 5 ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} ∈ V) → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
2624, 25mpan2 689 . . . 4 (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ V)
278, 26eqeltrid 2830 . . 3 (recs(𝐹) ∈ V → 𝐶 ∈ V)
28 fneq1 6651 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
29 fveq1 6900 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
30 reseq1 5983 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
3130fveq2d 6905 . . . . . . . 8 (𝑓 = 𝐶 → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐶𝑦)))
3229, 31eqeq12d 2742 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3332ralbidv 3168 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦))))
3428, 33anbi12d 630 . . . . 5 (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3534rexbidv 3169 . . . 4 (𝑓 = 𝐶 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3635, 1elab2g 3668 . . 3 (𝐶 ∈ V → (𝐶𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3727, 36syl 17 . 2 (recs(𝐹) ∈ V → (𝐶𝐴 ↔ ∃𝑥 ∈ On (𝐶 Fn 𝑥 ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐹‘(𝐶𝑦)))))
3823, 37mpbird 256 1 (recs(𝐹) ∈ V → 𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  {cab 2703  wral 3051  wrex 3060  Vcvv 3462  cun 3945  {csn 4633  cop 4639  dom cdm 5682  cres 5684  Ord word 6375  Oncon0 6376  suc csuc 6378   Fn wfn 6549  cfv 6554  recscrecs 8400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fo 6560  df-fv 6562  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401
This theorem is referenced by:  tfrlem13  8420
  Copyright terms: Public domain W3C validator