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

Theorem tfrlem12 8389
Description: Lemma for transfinite recursion. Show 𝐢 is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
tfrlem.3 𝐢 = (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
Assertion
Ref Expression
tfrlem12 (recs(𝐹) ∈ V β†’ 𝐢 ∈ 𝐴)
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐢   𝑓,𝐹,π‘₯,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑓)

Proof of Theorem tfrlem12
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . . . 6 𝐴 = {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
21tfrlem8 8384 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V β†’ Ord dom recs(𝐹))
4 dmexg 7894 . . . 4 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ V)
5 elon2 6376 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 584 . . 3 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ On)
7 onsuc 7799 . . . 4 (dom recs(𝐹) ∈ On β†’ suc dom recs(𝐹) ∈ On)
8 tfrlem.3 . . . . 5 𝐢 = (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
91, 8tfrlem10 8387 . . . 4 (dom recs(𝐹) ∈ On β†’ 𝐢 Fn suc dom recs(𝐹))
101, 8tfrlem11 8388 . . . . . 6 (dom recs(𝐹) ∈ On β†’ (𝑧 ∈ suc dom recs(𝐹) β†’ (πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧))))
1110ralrimiv 3146 . . . . 5 (dom recs(𝐹) ∈ On β†’ βˆ€π‘§ ∈ suc dom recs(𝐹)(πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)))
12 fveq2 6892 . . . . . . 7 (𝑧 = 𝑦 β†’ (πΆβ€˜π‘§) = (πΆβ€˜π‘¦))
13 reseq2 5977 . . . . . . . 8 (𝑧 = 𝑦 β†’ (𝐢 β†Ύ 𝑧) = (𝐢 β†Ύ 𝑦))
1413fveq2d 6896 . . . . . . 7 (𝑧 = 𝑦 β†’ (πΉβ€˜(𝐢 β†Ύ 𝑧)) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
1512, 14eqeq12d 2749 . . . . . 6 (𝑧 = 𝑦 β†’ ((πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)) ↔ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
1615cbvralvw 3235 . . . . 5 (βˆ€π‘§ ∈ suc dom recs(𝐹)(πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)) ↔ βˆ€π‘¦ ∈ suc dom recs(𝐹)(πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
1711, 16sylib 217 . . . 4 (dom recs(𝐹) ∈ On β†’ βˆ€π‘¦ ∈ suc dom recs(𝐹)(πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
18 fneq2 6642 . . . . . 6 (π‘₯ = suc dom recs(𝐹) β†’ (𝐢 Fn π‘₯ ↔ 𝐢 Fn suc dom recs(𝐹)))
19 raleq 3323 . . . . . 6 (π‘₯ = suc dom recs(𝐹) β†’ (βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)) ↔ βˆ€π‘¦ ∈ suc dom recs(𝐹)(πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
2018, 19anbi12d 632 . . . . 5 (π‘₯ = suc dom recs(𝐹) β†’ ((𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))) ↔ (𝐢 Fn suc dom recs(𝐹) ∧ βˆ€π‘¦ ∈ suc dom recs(𝐹)(πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))))
2120rspcev 3613 . . . 4 ((suc dom recs(𝐹) ∈ On ∧ (𝐢 Fn suc dom recs(𝐹) ∧ βˆ€π‘¦ ∈ suc dom recs(𝐹)(πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))) β†’ βˆƒπ‘₯ ∈ On (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
227, 9, 17, 21syl12anc 836 . . 3 (dom recs(𝐹) ∈ On β†’ βˆƒπ‘₯ ∈ On (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
236, 22syl 17 . 2 (recs(𝐹) ∈ V β†’ βˆƒπ‘₯ ∈ On (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
24 snex 5432 . . . . 5 {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩} ∈ V
25 unexg 7736 . . . . 5 ((recs(𝐹) ∈ V ∧ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩} ∈ V) β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ V)
2624, 25mpan2 690 . . . 4 (recs(𝐹) ∈ V β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ V)
278, 26eqeltrid 2838 . . 3 (recs(𝐹) ∈ V β†’ 𝐢 ∈ V)
28 fneq1 6641 . . . . . 6 (𝑓 = 𝐢 β†’ (𝑓 Fn π‘₯ ↔ 𝐢 Fn π‘₯))
29 fveq1 6891 . . . . . . . 8 (𝑓 = 𝐢 β†’ (π‘“β€˜π‘¦) = (πΆβ€˜π‘¦))
30 reseq1 5976 . . . . . . . . 9 (𝑓 = 𝐢 β†’ (𝑓 β†Ύ 𝑦) = (𝐢 β†Ύ 𝑦))
3130fveq2d 6896 . . . . . . . 8 (𝑓 = 𝐢 β†’ (πΉβ€˜(𝑓 β†Ύ 𝑦)) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
3229, 31eqeq12d 2749 . . . . . . 7 (𝑓 = 𝐢 β†’ ((π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
3332ralbidv 3178 . . . . . 6 (𝑓 = 𝐢 β†’ (βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
3428, 33anbi12d 632 . . . . 5 (𝑓 = 𝐢 β†’ ((𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))) ↔ (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))))
3534rexbidv 3179 . . . 4 (𝑓 = 𝐢 β†’ (βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦))) ↔ βˆƒπ‘₯ ∈ On (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))))
3635, 1elab2g 3671 . . 3 (𝐢 ∈ V β†’ (𝐢 ∈ 𝐴 ↔ βˆƒπ‘₯ ∈ On (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))))
3727, 36syl 17 . 2 (recs(𝐹) ∈ V β†’ (𝐢 ∈ 𝐴 ↔ βˆƒπ‘₯ ∈ On (𝐢 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))))
3823, 37mpbird 257 1 (recs(𝐹) ∈ V β†’ 𝐢 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947  {csn 4629  βŸ¨cop 4635  dom cdm 5677   β†Ύ cres 5679  Ord word 6364  Oncon0 6365  suc csuc 6367   Fn wfn 6539  β€˜cfv 6544  recscrecs 8370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371
This theorem is referenced by:  tfrlem13  8390
  Copyright terms: Public domain W3C validator