Theorem tfrlem7 6222

Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem7	⊢ Fun recs(𝐹)

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem7

Dummy variables 𝑔 ℎ 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . 3 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem6 6221	. 2 ⊢ Rel recs(𝐹)
3	1	recsfval 6220	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
4	3	eleq2i 2207	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐴)
5		eluni 3747	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
6	4, 5	bitri 183	. . . . . . 7 ⊢ (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
7	3	eleq2i 2207	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐴)
8		eluni 3747	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐴 ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴))
9	7, 8	bitri 183	. . . . . . 7 ⊢ (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴))
10	6, 9	anbi12i 456	. . . . . 6 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
11		eeanv 1905	. . . . . 6 ⊢ (∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
12	10, 11	bitr4i 186	. . . . 5 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
13		df-br 3938	. . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14		df-br 3938	. . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
15	13, 14	anbi12i 456	. . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ℎ))
16	1	tfrlem5 6219	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
17	16	impcom 124	. . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
18	15, 17	sylanbr 283	. . . . . . 7 ⊢ (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ℎ) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
19	18	an4s 578	. . . . . 6 ⊢ (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
20	19	exlimivv 1869	. . . . 5 ⊢ (∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
21	12, 20	sylbi 120	. . . 4 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
22	21	ax-gen 1426	. . 3 ⊢ ∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
23	22	gen2 1427	. 2 ⊢ ∀𝑥∀𝑢∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24		dffun4 5142	. 2 ⊢ (Fun recs(𝐹) ↔ (Rel recs(𝐹) ∧ ∀𝑥∀𝑢∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)))
25	2, 23, 24	mpbir2an 927	1 ⊢ Fun recs(𝐹)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  ⟨cop 3535  ∪ cuni 3744   class class class wbr 3937  Oncon0 4293   ↾ cres 4549  Rel wrel 4552  Fun wfun 5125   Fn wfn 5126  ‘cfv 5131  recscrecs 6209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-recs 6210

This theorem is referenced by:  tfrlem9  6224  tfrfun  6225  tfrlemibfn  6233  tfrlemiubacc  6235  tfri1d  6240  rdgfun  6278