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Theorem tfrlem12 8391
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 8386 . . . . 5 Ord dom recs(𝐹)
32a1i 11 . . . 4 (recs(𝐹) ∈ V β†’ Ord dom recs(𝐹))
4 dmexg 7896 . . . 4 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ V)
5 elon2 6374 . . . 4 (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
63, 4, 5sylanbrc 581 . . 3 (recs(𝐹) ∈ V β†’ dom recs(𝐹) ∈ On)
7 onsuc 7801 . . . 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 8389 . . . 4 (dom recs(𝐹) ∈ On β†’ 𝐢 Fn suc dom recs(𝐹))
101, 8tfrlem11 8390 . . . . . 6 (dom recs(𝐹) ∈ On β†’ (𝑧 ∈ suc dom recs(𝐹) β†’ (πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧))))
1110ralrimiv 3143 . . . . 5 (dom recs(𝐹) ∈ On β†’ βˆ€π‘§ ∈ suc dom recs(𝐹)(πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)))
12 fveq2 6890 . . . . . . 7 (𝑧 = 𝑦 β†’ (πΆβ€˜π‘§) = (πΆβ€˜π‘¦))
13 reseq2 5975 . . . . . . . 8 (𝑧 = 𝑦 β†’ (𝐢 β†Ύ 𝑧) = (𝐢 β†Ύ 𝑦))
1413fveq2d 6894 . . . . . . 7 (𝑧 = 𝑦 β†’ (πΉβ€˜(𝐢 β†Ύ 𝑧)) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
1512, 14eqeq12d 2746 . . . . . 6 (𝑧 = 𝑦 β†’ ((πΆβ€˜π‘§) = (πΉβ€˜(𝐢 β†Ύ 𝑧)) ↔ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
1615cbvralvw 3232 . . . . 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 6640 . . . . . 6 (π‘₯ = suc dom recs(𝐹) β†’ (𝐢 Fn π‘₯ ↔ 𝐢 Fn suc dom recs(𝐹)))
19 raleq 3320 . . . . . 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 629 . . . . 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 3611 . . . 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 833 . . 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 5430 . . . . 5 {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩} ∈ V
25 unexg 7738 . . . . 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 687 . . . 4 (recs(𝐹) ∈ V β†’ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}) ∈ V)
278, 26eqeltrid 2835 . . 3 (recs(𝐹) ∈ V β†’ 𝐢 ∈ V)
28 fneq1 6639 . . . . . 6 (𝑓 = 𝐢 β†’ (𝑓 Fn π‘₯ ↔ 𝐢 Fn π‘₯))
29 fveq1 6889 . . . . . . . 8 (𝑓 = 𝐢 β†’ (π‘“β€˜π‘¦) = (πΆβ€˜π‘¦))
30 reseq1 5974 . . . . . . . . 9 (𝑓 = 𝐢 β†’ (𝑓 β†Ύ 𝑦) = (𝐢 β†Ύ 𝑦))
3130fveq2d 6894 . . . . . . . 8 (𝑓 = 𝐢 β†’ (πΉβ€˜(𝑓 β†Ύ 𝑦)) = (πΉβ€˜(𝐢 β†Ύ 𝑦)))
3229, 31eqeq12d 2746 . . . . . . 7 (𝑓 = 𝐢 β†’ ((π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)) ↔ (πΆβ€˜π‘¦) = (πΉβ€˜(𝐢 β†Ύ 𝑦))))
3332ralbidv 3175 . . . . . 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 629 . . . . 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 3176 . . . 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 3669 . . 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 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆͺ cun 3945  {csn 4627  βŸ¨cop 4633  dom cdm 5675   β†Ύ cres 5677  Ord word 6362  Oncon0 6363  suc csuc 6365   Fn wfn 6537  β€˜cfv 6542  recscrecs 8372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373
This theorem is referenced by:  tfrlem13  8392
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