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Theorem dfrecs3OLD 8319
Description: Obsolete proof of dfrecs3 8318 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 8317 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
2 dfwrecsOLD 8244 . 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 1095 . . . . . . 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 3449 . . . . . . . . . . . 12 𝑥 ∈ V
54elon 6326 . . . . . . . . . . 11 (𝑥 ∈ On ↔ Ord 𝑥)
6 ordsson 7717 . . . . . . . . . . . . 13 (Ord 𝑥𝑥 ⊆ On)
7 ordtr 6331 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
86, 7jca 512 . . . . . . . . . . . 12 (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
9 epweon 7709 . . . . . . . . . . . . . . 15 E We On
10 wess 5620 . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . 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 5228 . . . . . . . . . . . . 13 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
17 ssel2 3939 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
18 predon 7720 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
1918sseq1d 3975 . . . . . . . . . . . . . . 15 (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2017, 19syl 17 . . . . . . . . . . . . . 14 ((𝑥 ⊆ On ∧ 𝑦𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥𝑦𝑥))
2120ralbidva 3172 . . . . . . . . . . . . 13 (𝑥 ⊆ On → (∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥))
2216, 21bitr4id 289 . . . . . . . . . . . 12 (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
2322pm5.32i 575 . . . . . . . . . . 11 ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
245, 15, 233bitri 296 . . . . . . . . . 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 6342 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
2718reseq2d 5937 . . . . . . . . . . . . . 14 (𝑦 ∈ On → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓𝑦))
2827fveq2d 6846 . . . . . . . . . . . . 13 (𝑦 ∈ On → (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓𝑦)))
2928eqeq2d 2747 . . . . . . . . . . . 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 3172 . . . . . . . . . 10 (𝑥 ∈ On → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3231pm5.32i 575 . . . . . . . . 9 ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
3325, 32bitr3i 276 . . . . . . . 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 643 . . . . . . 7 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
363, 34, 353bitri 296 . . . . . 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 1850 . . . . 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 3074 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
3937, 38bitr4i 277 . . . 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 2806 . . 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 4878 . 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 2768 1 recs(𝐹) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  wss 3910   cuni 4865  Tr wtr 5222   E cep 5536   We wwe 5587  cres 5635  Predcpred 6252  Ord word 6316  Oncon0 6317   Fn wfn 6491  cfv 6496  wrecscwrecs 8242  recscrecs 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317
This theorem is referenced by: (None)
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