Theorem tfrlem7 6375

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 6374	. 2 ⊢ Rel recs(𝐹)
3	1	recsfval 6373	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
4	3	eleq2i 2263	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐴)
5		eluni 3842	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
6	4, 5	bitri 184	. . . . . . 7 ⊢ (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
7	3	eleq2i 2263	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐴)
8		eluni 3842	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐴 ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴))
9	7, 8	bitri 184	. . . . . . 7 ⊢ (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴))
10	6, 9	anbi12i 460	. . . . . 6 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
11		eeanv 1951	. . . . . 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 187	. . . . 5 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
13		df-br 4034	. . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14		df-br 4034	. . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
15	13, 14	anbi12i 460	. . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ℎ))
16	1	tfrlem5 6372	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
17	16	impcom 125	. . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
18	15, 17	sylanbr 285	. . . . . . 7 ⊢ (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ℎ) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
19	18	an4s 588	. . . . . 6 ⊢ (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
20	19	exlimivv 1911	. . . . 5 ⊢ (∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
21	12, 20	sylbi 121	. . . 4 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
22	21	ax-gen 1463	. . 3 ⊢ ∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
23	22	gen2 1464	. 2 ⊢ ∀𝑥∀𝑢∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24		dffun4 5269	. 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 944	1 ⊢ Fun recs(𝐹)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  ⟨cop 3625  ∪ cuni 3839   class class class wbr 4033  Oncon0 4398   ↾ cres 4665  Rel wrel 4668  Fun wfun 5252   Fn wfn 5253  ‘cfv 5258  recscrecs 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-recs 6363

This theorem is referenced by:  tfrlem9  6377  tfrfun  6378  tfrlemibfn  6386  tfrlemiubacc  6388  tfri1d  6393  rdgfun  6431