Theorem lrrecfr 28329

Description: Now we show that 𝑅 is founded over No. (Contributed by Scott Fenton, 19-Aug-2024.)

Hypothesis

Ref	Expression
lrrec.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}

Assertion

Ref	Expression
lrrecfr	⊢ 𝑅 Fr No

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝑅(𝑥, 𝑦)

Proof of Theorem lrrecfr

Dummy variables 𝑎 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fr 5603	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 28133	. . . . 5 ⊢ Fun bday
3		imassrn 6197	. . . . . . 7 ⊢ (bday “ 𝑎) ⊆ ran bday
4		bdayrn 28137	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 3956	. . . . . 6 ⊢ (bday “ 𝑎) ⊆ On
6		fvex 6898	. . . . . . . . . . . . 13 ⊢ (bday‘𝑞) ∈ V
7	6	jctr 534	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ (bday‘𝑞) ∈ V))
8	7	eximi 1867	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ (bday‘𝑞) ∈ V))
9		n0 4299	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3087	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 (bday‘𝑞) ∈ V ↔ ∃𝑞(𝑞 ∈ 𝑎 ∧ (bday‘𝑞) ∈ V))
11	8, 9, 10	3imtr4i 295	. . . . . . . . . 10 ⊢ (𝑎 ≠ ∅ → ∃𝑞 ∈ 𝑎 (bday‘𝑞) ∈ V)
12		isset 3464	. . . . . . . . . . . . 13 ⊢ ((bday‘𝑞) ∈ V ↔ ∃𝑝 𝑝 = (bday‘𝑞))
13		eqcom 2767	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (bday‘𝑞) ↔ (bday‘𝑞) = 𝑝)
14	13	exbii 1880	. . . . . . . . . . . . 13 ⊢ (∃𝑝 𝑝 = (bday‘𝑞) ↔ ∃𝑝(bday‘𝑞) = 𝑝)
15	12, 14	bitri 278	. . . . . . . . . . . 12 ⊢ ((bday‘𝑞) ∈ V ↔ ∃𝑝(bday‘𝑞) = 𝑝)
16	15	rexbii 3109	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 (bday‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝(bday‘𝑞) = 𝑝)
17		rexcom4 3289	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ∃𝑝(bday‘𝑞) = 𝑝 ↔ ∃𝑝∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝)
18	16, 17	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ 𝑎 (bday‘𝑞) ∈ V ↔ ∃𝑝∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝)
19	11, 18	sylib 221	. . . . . . . . 9 ⊢ (𝑎 ≠ ∅ → ∃𝑝∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝)
20	19	adantl 487	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝)
21		bdayfn 28134	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6957	. . . . . . . . . . 11 ⊢ ((bday Fn No ∧ 𝑎 ⊆ No) → (𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
23	21, 22	mpan 703	. . . . . . . . . 10 ⊢ (𝑎 ⊆ No → (𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
24	23	adantr 486	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
25	24	exbidv 1953	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑝∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
26	20, 25	mpbird 260	. . . . . . 7 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 𝑝 ∈ (bday “ 𝑎))
27		n0 4299	. . . . . . 7 ⊢ ((bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (bday “ 𝑎))
28	26, 27	sylibr 237	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (bday “ 𝑎) ≠ ∅)
29		onint 7804	. . . . . 6 ⊢ (((bday “ 𝑎) ⊆ On ∧ (bday “ 𝑎) ≠ ∅) → ∩ (bday “ 𝑎) ∈ (bday “ 𝑎))
30	5, 28, 29	sylancr 599	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∩ (bday “ 𝑎) ∈ (bday “ 𝑎))
31		fvelima 6950	. . . . 5 ⊢ ((Fun bday ∧ ∩ (bday “ 𝑎) ∈ (bday “ 𝑎)) → ∃𝑝 ∈ 𝑎 (bday‘𝑝) = ∩ (bday “ 𝑎))
32	2, 30, 31	sylancr 599	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 (bday‘𝑝) = ∩ (bday “ 𝑎))
33		fnfvima 7239	. . . . . . . . . 10 ⊢ ((bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → (bday‘𝑞) ∈ (bday “ 𝑎))
34	21, 33	mp3an1 1477	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → (bday‘𝑞) ∈ (bday “ 𝑎))
35	34	adantlr 728	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → (bday‘𝑞) ∈ (bday “ 𝑎))
36		onnmin 7812	. . . . . . . 8 ⊢ (((bday “ 𝑎) ⊆ On ∧ (bday‘𝑞) ∈ (bday “ 𝑎)) → ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎))
37	5, 35, 36	sylancr 599	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎))
38	37	ralrimiva 3154	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎))
39		eleq2 2849	. . . . . . . 8 ⊢ ((bday‘𝑝) = ∩ (bday “ 𝑎) → ((bday‘𝑞) ∈ (bday‘𝑝) ↔ (bday‘𝑞) ∈ ∩ (bday “ 𝑎)))
40	39	notbid 321	. . . . . . 7 ⊢ ((bday‘𝑝) = ∩ (bday “ 𝑎) → (¬ (bday‘𝑞) ∈ (bday‘𝑝) ↔ ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎)))
41	40	ralbidv 3185	. . . . . 6 ⊢ ((bday‘𝑝) = ∩ (bday “ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝) ↔ ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎)))
42	38, 41	syl5ibrcom 250	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ((bday‘𝑝) = ∩ (bday “ 𝑎) → ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
43	42	reximdv 3177	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 ∈ 𝑎 (bday‘𝑝) = ∩ (bday “ 𝑎) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
44	32, 43	mpd 16	. . 3 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝))
45		simpll 779	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No)
46		simprr 785	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 3909	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No)
48		simprl 783	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 3909	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ No)
50		lrrec.1	. . . . . . . . 9 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ ((L‘𝑦) ∪ (R‘𝑦))}
51	50	lrrecval2 28326	. . . . . . . 8 ⊢ ((𝑞 ∈ No ∧ 𝑝 ∈ No) → (𝑞𝑅𝑝 ↔ (bday‘𝑞) ∈ (bday‘𝑝)))
52	47, 49, 51	syl2anc 596	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → (𝑞𝑅𝑝 ↔ (bday‘𝑞) ∈ (bday‘𝑝)))
53	52	notbid 321	. . . . . 6 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → (¬ 𝑞𝑅𝑝 ↔ ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
54	53	anassrs 473	. . . . 5 ⊢ ((((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) ∧ 𝑞 ∈ 𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
55	54	ralbidva 3183	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
56	55	rexbidva 3184	. . 3 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
57	44, 56	mpbird 260	. 2 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝)
58	1, 57	mpgbir 1831	1 ⊢ 𝑅 Fr No

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  ∩ cint 4906   class class class wbr 5102  {copab 5166   Fr wfr 5600  ran crn 5651   “ cima 5653  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  Nocsur 27997  bdaycbday 27999  Lcleft 28211  Rcright 28212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217

This theorem is used by:  noinds  28331  norecfn  28332  norecov  28333  noxpordfr  28337  no2indlesm  28340  no3inds  28344