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Theorem dfrecs3OLD 8412
Description: Obsolete version of dfrecs3 8411 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 3-Aug-2020.)
Assertion
Ref Expression
dfrecs3OLD recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Distinct variable group:   𝑓,𝐹,𝑥,𝑦

Proof of Theorem dfrecs3OLD
StepHypRef Expression
1 df-recs 8410 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
2 dfwrecsOLD 8337 . 2 wrecs( E , On, 𝐹) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))}
3 3anass 1094 . . . . . . 7 ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))))
4 vex 3482 . . . . . . . . . . . 12 𝑥 ∈ V
54elon 6395 . . . . . . . . . . 11 (𝑥 ∈ On ↔ Ord 𝑥)
6 ordsson 7802 . . . . . . . . . . . . 13 (Ord 𝑥𝑥 ⊆ On)
7 ordtr 6400 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
86, 7jca 511 . . . . . . . . . . . 12 (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
9 epweon 7794 . . . . . . . . . . . . . . 15 E We On
10 wess 5675 . . . . . . . . . . . . . . 15 (𝑥 ⊆ On → ( E We On → E We 𝑥))
119, 10mpi 20 . . . . . . . . . . . . . 14 (𝑥 ⊆ On → E We 𝑥)
1211anim1ci 616 . . . . . . . . . . . . 13 ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
13 df-ord 6389 . . . . . . . . . . . . 13 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
1412, 13sylibr 234 . . . . . . . . . . . 12 ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
158, 14impbii 209 . . . . . . . . . . 11 (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
16 dftr3 5271 . . . . . . . . . . . . 13 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
17 ssel2 3990 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
18 predon 7805 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
1918sseq1d 4027 . . . . . . . . . . . . . . 15 (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2017, 19syl 17 . . . . . . . . . . . . . 14 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2120ralbidva 3174 . . . . . . . . . . . . 13 (𝑥 ⊆ On → (∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥))
2216, 21bitr4id 290 . . . . . . . . . . . 12 (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
2322pm5.32i 574 . . . . . . . . . . 11 ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
245, 15, 233bitri 297 . . . . . . . . . 10 (𝑥 ∈ On ↔ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
2524anbi1i 624 . . . . . . . . 9 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))))
26 onelon 6411 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2718reseq2d 6000 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓𝑦))
2827fveq2d 6911 . . . . . . . . . . . . 13 (𝑦 ∈ On → (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓𝑦)))
2928eqeq2d 2746 . . . . . . . . . . . 12 (𝑦 ∈ On → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3026, 29syl 17 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦𝑥) → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3130ralbidva 3174 . . . . . . . . . 10 (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3231pm5.32i 574 . . . . . . . . 9 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3325, 32bitr3i 277 . . . . . . . 8 (((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3433anbi2i 623 . . . . . . 7 ((𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
35 an12 645 . . . . . . 7 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
363, 34, 353bitri 297 . . . . . 6 ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
3736exbii 1845 . . . . 5 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
38 df-rex 3069 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
3937, 38bitr4i 278 . . . 4 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
4039abbii 2807 . . 3 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
4140unieqi 4924 . 2 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
421, 2, 413eqtri 2767 1 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963   cuni 4912  Tr wtr 5265   E cep 5588   We wwe 5640  cres 5691  Predcpred 6322  Ord word 6385  Oncon0 6386   Fn wfn 6558  cfv 6563  wrecscwrecs 8335  recscrecs 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410
This theorem is referenced by: (None)
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