MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem11 Structured version   Visualization version   GIF version
Theorem tfrlem11 8390
Description: Lemma for transfinite recursion. Compute the value of 𝐢. (Contributed by NM, 18-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
tfrlem11 (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ suc dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐡   𝐢,𝑓,π‘₯,𝑦   𝑓,𝐹,π‘₯,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑓)
Proof of Theorem tfrlem11
StepHypRef Expression
1 elsuci 6430 . 2 (𝐡 ∈ suc dom recs(𝐹) β†’ (𝐡 ∈ dom recs(𝐹) ∨ 𝐡 = dom recs(𝐹)))
2 tfrlem.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ βˆƒπ‘₯ ∈ On (𝑓 Fn π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (πΉβ€˜(𝑓 β†Ύ 𝑦)))}
3 tfrlem.3 . . . . . . . . 9 𝐢 = (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
42, 3tfrlem10 8389 . . . . . . . 8 (dom recs(𝐹) ∈ On β†’ 𝐢 Fn suc dom recs(𝐹))
5 fnfun 6648 . . . . . . . 8 (𝐢 Fn suc dom recs(𝐹) β†’ Fun 𝐢)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On β†’ Fun 𝐢)
7 ssun1 4171 . . . . . . . . 9 recs(𝐹) βŠ† (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
87, 3sseqtrri 4018 . . . . . . . 8 recs(𝐹) βŠ† 𝐢
92tfrlem9 8387 . . . . . . . . 9 (𝐡 ∈ dom recs(𝐹) β†’ (recs(𝐹)β€˜π΅) = (πΉβ€˜(recs(𝐹) β†Ύ 𝐡)))
10 funssfv 6911 . . . . . . . . . . . 12 ((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢 ∧ 𝐡 ∈ dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (recs(𝐹)β€˜π΅))
11103expa 1116 . . . . . . . . . . 11 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ 𝐡 ∈ dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (recs(𝐹)β€˜π΅))
1211adantrl 712 . . . . . . . . . 10 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ (dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹))) β†’ (πΆβ€˜π΅) = (recs(𝐹)β€˜π΅))
13 onelss 6405 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ dom recs(𝐹) β†’ 𝐡 βŠ† dom recs(𝐹)))
1413imp 405 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹)) β†’ 𝐡 βŠ† dom recs(𝐹))
15 fun2ssres 6592 . . . . . . . . . . . . 13 ((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢 ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
16153expa 1116 . . . . . . . . . . . 12 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
1716fveq2d 6894 . . . . . . . . . . 11 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (πΉβ€˜(𝐢 β†Ύ 𝐡)) = (πΉβ€˜(recs(𝐹) β†Ύ 𝐡)))
1814, 17sylan2 591 . . . . . . . . . 10 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ (dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹))) β†’ (πΉβ€˜(𝐢 β†Ύ 𝐡)) = (πΉβ€˜(recs(𝐹) β†Ύ 𝐡)))
1912, 18eqeq12d 2746 . . . . . . . . 9 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ (dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹))) β†’ ((πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡)) ↔ (recs(𝐹)β€˜π΅) = (πΉβ€˜(recs(𝐹) β†Ύ 𝐡))))
209, 19imbitrrid 245 . . . . . . . 8 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ (dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹))) β†’ (𝐡 ∈ dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
218, 20mpanl2 697 . . . . . . 7 ((Fun 𝐢 ∧ (dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹))) β†’ (𝐡 ∈ dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
226, 21sylan 578 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ (dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹))) β†’ (𝐡 ∈ dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
2322exp32 419 . . . . 5 (dom recs(𝐹) ∈ On β†’ (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ dom recs(𝐹) β†’ (𝐡 ∈ dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))))
2423pm2.43i 52 . . . 4 (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ dom recs(𝐹) β†’ (𝐡 ∈ dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡)))))
2524pm2.43d 53 . . 3 (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
26 opex 5463 . . . . . . . . 9 ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ V
2726snid 4663 . . . . . . . 8 ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ {⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩}
28 opeq1 4872 . . . . . . . . . . 11 (𝐡 = dom recs(𝐹) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ = ⟨dom recs(𝐹), (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩)
2928adantl 480 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ = ⟨dom recs(𝐹), (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩)
30 eqimss 4039 . . . . . . . . . . . . . 14 (𝐡 = dom recs(𝐹) β†’ 𝐡 βŠ† dom recs(𝐹))
318, 15mp3an2 1447 . . . . . . . . . . . . . 14 ((Fun 𝐢 ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
326, 30, 31syl2an 594 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
33 reseq2 5975 . . . . . . . . . . . . . . 15 (𝐡 = dom recs(𝐹) β†’ (recs(𝐹) β†Ύ 𝐡) = (recs(𝐹) β†Ύ dom recs(𝐹)))
342tfrlem6 8384 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 6025 . . . . . . . . . . . . . . . 16 (Rel recs(𝐹) β†’ (recs(𝐹) β†Ύ dom recs(𝐹)) = recs(𝐹))
3634, 35ax-mp 5 . . . . . . . . . . . . . . 15 (recs(𝐹) β†Ύ dom recs(𝐹)) = recs(𝐹)
3733, 36eqtrdi 2786 . . . . . . . . . . . . . 14 (𝐡 = dom recs(𝐹) β†’ (recs(𝐹) β†Ύ 𝐡) = recs(𝐹))
3837adantl 480 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (recs(𝐹) β†Ύ 𝐡) = recs(𝐹))
3932, 38eqtrd 2770 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = recs(𝐹))
4039fveq2d 6894 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (πΉβ€˜(𝐢 β†Ύ 𝐡)) = (πΉβ€˜recs(𝐹)))
4140opeq2d 4879 . . . . . . . . . 10 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨dom recs(𝐹), (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ = ⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩)
4229, 41eqtrd 2770 . . . . . . . . 9 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ = ⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩)
4342sneqd 4639 . . . . . . . 8 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ {⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩} = {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
4427, 43eleqtrid 2837 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
45 elun2 4176 . . . . . . 7 (⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩} β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}))
4644, 45syl 17 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩}))
4746, 3eleqtrrdi 2842 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ 𝐢)
48 simpr 483 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ 𝐡 = dom recs(𝐹))
49 sucidg 6444 . . . . . . . 8 (dom recs(𝐹) ∈ On β†’ dom recs(𝐹) ∈ suc dom recs(𝐹))
5049adantr 479 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ dom recs(𝐹) ∈ suc dom recs(𝐹))
5148, 50eqeltrd 2831 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ 𝐡 ∈ suc dom recs(𝐹))
52 fnopfvb 6944 . . . . . 6 ((𝐢 Fn suc dom recs(𝐹) ∧ 𝐡 ∈ suc dom recs(𝐹)) β†’ ((πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡)) ↔ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ 𝐢))
534, 51, 52syl2an2r 681 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ((πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡)) ↔ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ 𝐢))
5447, 53mpbird 256 . . . 4 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡)))
5554ex 411 . . 3 (dom recs(𝐹) ∈ On β†’ (𝐡 = dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
5625, 55jaod 855 . 2 (dom recs(𝐹) ∈ On β†’ ((𝐡 ∈ dom recs(𝐹) ∨ 𝐡 = dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
571, 56syl5 34 1 (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ suc dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  βˆƒwrex 3068   βˆͺ cun 3945   βŠ† wss 3947  {csn 4627  βŸ¨cop 4633  dom cdm 5675   β†Ύ cres 5677  Rel wrel 5680  Oncon0 6363  suc csuc 6365  Fun wfun 6536   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:  tfrlem12  8391
  Copyright terms: Public domain W3C validator