Theorem tfrlem7 8354

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 8352	. 2 ⊢ Rel recs(𝐹)
3	1	recsfval 8351	. . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴
4	3	eleq2i 2854	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐴)
5		eluni 4868	. . . . . . . 8 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐴 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
6	4, 5	bitri 277	. . . . . . 7 ⊢ (⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴))
7	3	eleq2i 2854	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐴)
8		eluni 4868	. . . . . . . 8 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐴 ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴))
9	7, 8	bitri 277	. . . . . . 7 ⊢ (⟨𝑥, 𝑣⟩ ∈ recs(𝐹) ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴))
10	6, 9	anbi12i 637	. . . . . 6 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ (∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
11		exdistrv 1975	. . . . . 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 280	. . . . 5 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)))
13		df-br 5101	. . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
14		df-br 5101	. . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
15	13, 14	anbi12i 637	. . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ℎ))
16	1	tfrlem5 8350	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
17	16	impcom 411	. . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
18	15, 17	sylanbr 591	. . . . . . 7 ⊢ (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ ⟨𝑥, 𝑣⟩ ∈ ℎ) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
19	18	an4s 670	. . . . . 6 ⊢ (((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
20	19	exlimivv 1952	. . . . 5 ⊢ (∃𝑔∃ℎ((⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣)
21	12, 20	sylbi 219	. . . 4 ⊢ ((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
22	21	ax-gen 1815	. . 3 ⊢ ∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
23	22	gen2 1816	. 2 ⊢ ∀𝑥∀𝑢∀𝑣((⟨𝑥, 𝑢⟩ ∈ recs(𝐹) ∧ ⟨𝑥, 𝑣⟩ ∈ recs(𝐹)) → 𝑢 = 𝑣)
24		dffun4 6534	. 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 721	1 ⊢ Fun recs(𝐹)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1558   = wceq 1560  ∃wex 1799   ∈ wcel 2142  {cab 2740  ∀wral 3076  ∃wrex 3086  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100   ↾ cres 5649  Rel wrel 5652  Oncon0 6346  Fun wfun 6515   Fn wfn 6516  ‘cfv 6521  recscrecs 8341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fo 6527  df-fv 6529  df-ov 7399  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342

This theorem is referenced by:  tfrlem9  8356  tfrlem9a  8357  tfrlem10  8358  tfrlem14  8362  tfrlem16  8364  tfr1a  8365  tfr1  8368