Theorem dfrecs3OLD 8429

Description: Obsolete version of dfrecs3 8428 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Distinct variable group:   𝑓,𝐹,𝑥,𝑦

Proof of Theorem dfrecs3OLD

Step	Hyp	Ref	Expression
1		df-recs 8427	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2		dfwrecsOLD 8354	. 2 ⊢ wrecs( E , On, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))}
3		3anass 1095	. . . . . . 7 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))))
4		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
5	4	elon 6404	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
6		ordsson 7818	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → 𝑥 ⊆ On)
7		ordtr 6409	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → Tr 𝑥)
8	6, 7	jca 511	. . . . . . . . . . . 12 ⊢ (Ord 𝑥 → (𝑥 ⊆ On ∧ Tr 𝑥))
9		epweon 7810	. . . . . . . . . . . . . . 15 ⊢ E We On
10		wess 5686	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ On → ( E We On → E We 𝑥))
11	9, 10	mpi 20	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ On → E We 𝑥)
12	11	anim1ci 615	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → (Tr 𝑥 ∧ E We 𝑥))
13		df-ord 6398	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 ↔ (Tr 𝑥 ∧ E We 𝑥))
14	12, 13	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) → Ord 𝑥)
15	8, 14	impbii 209	. . . . . . . . . . 11 ⊢ (Ord 𝑥 ↔ (𝑥 ⊆ On ∧ Tr 𝑥))
16		dftr3 5289	. . . . . . . . . . . . 13 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
17		ssel2 4003	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
18		predon 7821	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → Pred( E , On, 𝑦) = 𝑦)
19	18	sseq1d 4040	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ On → (Pred( E , On, 𝑦) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
20	17, 19	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → (Pred( E , On, 𝑦) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥))
21	20	ralbidva 3182	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ On → (∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥))
22	16, 21	bitr4id 290	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ On → (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
23	22	pm5.32i 574	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ On ∧ Tr 𝑥) ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
24	5, 15, 23	3bitri 297	. . . . . . . . . 10 ⊢ (𝑥 ∈ On ↔ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥))
25	24	anbi1i 623	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))))
26		onelon 6420	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
27	18	reseq2d 6009	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ On → (𝑓 ↾ Pred( E , On, 𝑦)) = (𝑓 ↾ 𝑦))
28	27	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) = (𝐹‘(𝑓 ↾ 𝑦)))
29	28	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
30	26, 29	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
31	30	ralbidva 3182	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
32	31	pm5.32i 574	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
33	25, 32	bitr3i 277	. . . . . . . 8 ⊢ (((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
34	33	anbi2i 622	. . . . . . 7 ⊢ ((𝑓 Fn 𝑥 ∧ ((𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
35		an12 644	. . . . . . 7 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
36	3, 34, 35	3bitri 297	. . . . . 6 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ (𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
37	36	exbii 1846	. . . . 5 ⊢ (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
38		df-rex 3077	. . . . 5 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
39	37, 38	bitr4i 278	. . . 4 ⊢ (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦)))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
40	39	abbii 2812	. . 3 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
41	40	unieqi 4943	. 2 ⊢ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ On ∧ ∀𝑦 ∈ 𝑥 Pred( E , On, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred( E , On, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
42	1, 2, 41	3eqtri 2772	1 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∪ cuni 4931  Tr wtr 5283   E cep 5598   We wwe 5651   ↾ cres 5702  Predcpred 6331  Ord word 6394  Oncon0 6395   Fn wfn 6568  ‘cfv 6573  wrecscwrecs 8352  recscrecs 8426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427

This theorem is referenced by: (None)