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Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfrlem11 Structured version   Visualization version   GIF version
Theorem tfrlem11 8391
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 6431 . 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 8390 . . . . . . . 8 (dom recs(𝐹) ∈ On β†’ 𝐢 Fn suc dom recs(𝐹))
5 fnfun 6649 . . . . . . . 8 (𝐢 Fn suc dom recs(𝐹) β†’ Fun 𝐢)
64, 5syl 17 . . . . . . 7 (dom recs(𝐹) ∈ On β†’ Fun 𝐢)
7 ssun1 4172 . . . . . . . . 9 recs(𝐹) βŠ† (recs(𝐹) βˆͺ {⟨dom recs(𝐹), (πΉβ€˜recs(𝐹))⟩})
87, 3sseqtrri 4019 . . . . . . . 8 recs(𝐹) βŠ† 𝐢
92tfrlem9 8388 . . . . . . . . 9 (𝐡 ∈ dom recs(𝐹) β†’ (recs(𝐹)β€˜π΅) = (πΉβ€˜(recs(𝐹) β†Ύ 𝐡)))
10 funssfv 6912 . . . . . . . . . . . 12 ((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢 ∧ 𝐡 ∈ dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (recs(𝐹)β€˜π΅))
11103expa 1117 . . . . . . . . . . 11 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ 𝐡 ∈ dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (recs(𝐹)β€˜π΅))
1211adantrl 713 . . . . . . . . . 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 6406 . . . . . . . . . . . 12 (dom recs(𝐹) ∈ On β†’ (𝐡 ∈ dom recs(𝐹) β†’ 𝐡 βŠ† dom recs(𝐹)))
1413imp 406 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐡 ∈ dom recs(𝐹)) β†’ 𝐡 βŠ† dom recs(𝐹))
15 fun2ssres 6593 . . . . . . . . . . . . 13 ((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢 ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
16153expa 1117 . . . . . . . . . . . 12 (((Fun 𝐢 ∧ recs(𝐹) βŠ† 𝐢) ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
1716fveq2d 6895 . . . . . . . . . . 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 592 . . . . . . . . . 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 2747 . . . . . . . . 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 698 . . . . . . 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 579 . . . . . 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 420 . . . . 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 5464 . . . . . . . . 9 ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ V
2726snid 4664 . . . . . . . 8 ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ {⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩}
28 opeq1 4873 . . . . . . . . . . 11 (𝐡 = dom recs(𝐹) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ = ⟨dom recs(𝐹), (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩)
2928adantl 481 . . . . . . . . . 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 4040 . . . . . . . . . . . . . 14 (𝐡 = dom recs(𝐹) β†’ 𝐡 βŠ† dom recs(𝐹))
318, 15mp3an2 1448 . . . . . . . . . . . . . 14 ((Fun 𝐢 ∧ 𝐡 βŠ† dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
326, 30, 31syl2an 595 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = (recs(𝐹) β†Ύ 𝐡))
33 reseq2 5976 . . . . . . . . . . . . . . 15 (𝐡 = dom recs(𝐹) β†’ (recs(𝐹) β†Ύ 𝐡) = (recs(𝐹) β†Ύ dom recs(𝐹)))
342tfrlem6 8385 . . . . . . . . . . . . . . . 16 Rel recs(𝐹)
35 resdm 6026 . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . 14 (𝐡 = dom recs(𝐹) β†’ (recs(𝐹) β†Ύ 𝐡) = recs(𝐹))
3837adantl 481 . . . . . . . . . . . . 13 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (recs(𝐹) β†Ύ 𝐡) = recs(𝐹))
3932, 38eqtrd 2771 . . . . . . . . . . . 12 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (𝐢 β†Ύ 𝐡) = recs(𝐹))
4039fveq2d 6895 . . . . . . . . . . 11 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (πΉβ€˜(𝐢 β†Ύ 𝐡)) = (πΉβ€˜recs(𝐹)))
4140opeq2d 4880 . . . . . . . . . 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 2771 . . . . . . . . 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 4640 . . . . . . . 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 2838 . . . . . . 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 4177 . . . . . . 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 2843 . . . . 5 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ ⟨𝐡, (πΉβ€˜(𝐢 β†Ύ 𝐡))⟩ ∈ 𝐢)
48 simpr 484 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ 𝐡 = dom recs(𝐹))
49 sucidg 6445 . . . . . . . 8 (dom recs(𝐹) ∈ On β†’ dom recs(𝐹) ∈ suc dom recs(𝐹))
5049adantr 480 . . . . . . 7 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ dom recs(𝐹) ∈ suc dom recs(𝐹))
5148, 50eqeltrd 2832 . . . . . 6 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ 𝐡 ∈ suc dom recs(𝐹))
52 fnopfvb 6945 . . . . . 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 682 . . . . 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 257 . . . 4 ((dom recs(𝐹) ∈ On ∧ 𝐡 = dom recs(𝐹)) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡)))
5554ex 412 . . 3 (dom recs(𝐹) ∈ On β†’ (𝐡 = dom recs(𝐹) β†’ (πΆβ€˜π΅) = (πΉβ€˜(𝐢 β†Ύ 𝐡))))
5625, 55jaod 856 . 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 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  βˆƒwrex 3069   βˆͺ cun 3946   βŠ† wss 3948  {csn 4628  βŸ¨cop 4634  dom cdm 5676   β†Ύ cres 5678  Rel wrel 5681  Oncon0 6364  suc csuc 6366  Fun wfun 6537   Fn wfn 6538  β€˜cfv 6543  recscrecs 8373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374
This theorem is referenced by:  tfrlem12  8392
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