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Theorem tfrlem12 8388
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 8383 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V β†’ Ord dom recs(𝐹))
4 dmexg 7893 . . . 4 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ V)
5 elon2 6375 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 583 . . 3 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ On)
7 onsuc 7798 . . . 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 8386 . . . 4 (dom recs(𝐹) ∈ On β†’ 𝐢 Fn suc dom recs(𝐹))
101, 8tfrlem11 8387 . . . . . 6 (dom recs(𝐹) ∈ On β†’ (𝑧 ∈ suc dom recs(𝐹) β†’ (πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧))))
1110ralrimiv 3145 . . . . 5 (dom recs(𝐹) ∈ On β†’ βˆ€π‘§ ∈ suc dom recs(𝐹)(πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)))
12 fveq2 6891 . . . . . . 7 (𝑧 = 𝑦 β†’ (πΆβ€˜π‘§) = (πΆβ€˜π‘¦))
13 reseq2 5976 . . . . . . . 8 (𝑧 = 𝑦 β†’ (𝐢 β†Ύ 𝑧) = (𝐢 β†Ύ 𝑦))
1413fveq2d 6895 . . . . . . 7 (𝑧 = 𝑦 β†’ (πΉβ€˜(𝐢 β†Ύ 𝑧)) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
1512, 14eqeq12d 2748 . . . . . 6 (𝑧 = 𝑦 β†’ ((πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)) ↔ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
1615cbvralvw 3234 . . . . 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 6641 . . . . . 6 (π‘₯ = suc dom recs(𝐹) β†’ (𝐢 Fn π‘₯ ↔ 𝐢 Fn suc dom recs(𝐹)))
19 raleq 3322 . . . . . 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 631 . . . . 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 3612 . . . 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 835 . . 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 5431 . . . . 5 {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩} ∈ V
25 unexg 7735 . . . . 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 689 . . . 4 (recs(𝐹) ∈ V β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ V)
278, 26eqeltrid 2837 . . 3 (recs(𝐹) ∈ V β†’ 𝐢 ∈ V)
28 fneq1 6640 . . . . . 6 (𝑓 = 𝐢 β†’ (𝑓 Fn π‘₯ ↔ 𝐢 Fn π‘₯))
29 fveq1 6890 . . . . . . . 8 (𝑓 = 𝐢 β†’ (π‘“β€˜π‘¦) = (πΆβ€˜π‘¦))
30 reseq1 5975 . . . . . . . . 9 (𝑓 = 𝐢 β†’ (𝑓 β†Ύ 𝑦) = (𝐢 β†Ύ 𝑦))
3130fveq2d 6895 . . . . . . . 8 (𝑓 = 𝐢 β†’ (πΉβ€˜(𝑓 β†Ύ 𝑦)) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
3229, 31eqeq12d 2748 . . . . . . 7 (𝑓 = 𝐢 β†’ ((π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
3332ralbidv 3177 . . . . . 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 631 . . . . 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 3178 . . . 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 3670 . . 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 256 1 (recs(𝐹) ∈ V β†’ 𝐢 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆͺ cun 3946  {csn 4628  βŸ¨cop 4634  dom cdm 5676   β†Ύ cres 5678  Ord word 6363  Oncon0 6364  suc csuc 6366   Fn wfn 6538  β€˜cfv 6543  recscrecs 8369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7411  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370
This theorem is referenced by:  tfrlem13  8389
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