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Theorem dfrecs3OLD 8374
Description: Obsolete version of dfrecs3 8373 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 8372 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
2 dfwrecsOLD 8299 . 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 1092 . . . . . . 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 3472 . . . . . . . . . . . 12 𝑥 ∈ V
54elon 6367 . . . . . . . . . . 11 (𝑥 ∈ On ↔ Ord 𝑥)
6 ordsson 7767 . . . . . . . . . . . . 13 (Ord 𝑥𝑥 ⊆ On)
7 ordtr 6372 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
86, 7jca 511 . . . . . . . . . . . 12 (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
9 epweon 7759 . . . . . . . . . . . . . . 15 E We On
10 wess 5656 . . . . . . . . . . . . . . 15 (𝑥 ⊆ On → ( E We On → E We 𝑥))
119, 10mpi 20 . . . . . . . . . . . . . 14 (𝑥 ⊆ On → E We 𝑥)
1211anim1ci 615 . . . . . . . . . . . . 13 ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
13 df-ord 6361 . . . . . . . . . . . . 13 (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
1412, 13sylibr 233 . . . . . . . . . . . 12 ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
158, 14impbii 208 . . . . . . . . . . 11 (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
16 dftr3 5264 . . . . . . . . . . . . 13 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
17 ssel2 3972 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
18 predon 7770 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
1918sseq1d 4008 . . . . . . . . . . . . . . 15 (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2017, 19syl 17 . . . . . . . . . . . . . 14 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2120ralbidva 3169 . . . . . . . . . . . . 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 623 . . . . . . . . 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 6383 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2718reseq2d 5975 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓𝑦))
2827fveq2d 6889 . . . . . . . . . . . . 13 (𝑦 ∈ On → (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓𝑦)))
2928eqeq2d 2737 . . . . . . . . . . . 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 3169 . . . . . . . . . 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 622 . . . . . . 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 642 . . . . . . 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 1842 . . . . 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 3065 . . . . 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 2796 . . 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 4914 . 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 2758 1 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wral 3055  wrex 3064  wss 3943   cuni 4902  Tr wtr 5258   E cep 5572   We wwe 5623  cres 5671  Predcpred 6293  Ord word 6357  Oncon0 6358   Fn wfn 6532  cfv 6537  wrecscwrecs 8297  recscrecs 8371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372
This theorem is referenced by: (None)
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