Theorem lrrecfr 27994

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 5652	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 27835	. . . . 5 ⊢ Fun bday
3		imassrn 6100	. . . . . . 7 ⊢ ( bday “ 𝑎) ⊆ ran bday
4		bdayrn 27838	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 4045	. . . . . 6 ⊢ ( bday “ 𝑎) ⊆ On
6		fvex 6933	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑞) ∈ V
7	6	jctr 524	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
8	7	eximi 1833	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
9		n0 4376	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3077	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
11	8, 9, 10	3imtr4i 292	. . . . . . . . . 10 ⊢ (𝑎 ≠ ∅ → ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V)
12		isset 3502	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday ‘𝑞))
13		eqcom 2747	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ( bday ‘𝑞) ↔ ( bday ‘𝑞) = 𝑝)
14	13	exbii 1846	. . . . . . . . . . . . 13 ⊢ (∃𝑝 𝑝 = ( bday ‘𝑞) ↔ ∃𝑝( bday ‘𝑞) = 𝑝)
15	12, 14	bitri 275	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝( bday ‘𝑞) = 𝑝)
16	15	rexbii 3100	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝)
17		rexcom4 3294	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝 ↔ ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
18	16, 17	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
19	11, 18	sylib 218	. . . . . . . . 9 ⊢ (𝑎 ≠ ∅ → ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
21		bdayfn 27836	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6994	. . . . . . . . . . 11 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
23	21, 22	mpan 689	. . . . . . . . . 10 ⊢ (𝑎 ⊆ No → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
24	23	adantr 480	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
25	24	exbidv 1920	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
26	20, 25	mpbird 257	. . . . . . 7 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
27		n0 4376	. . . . . . 7 ⊢ (( bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
28	26, 27	sylibr 234	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ( bday “ 𝑎) ≠ ∅)
29		onint 7826	. . . . . 6 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday “ 𝑎) ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
30	5, 28, 29	sylancr 586	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
31		fvelima 6987	. . . . 5 ⊢ ((Fun bday ∧ ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎)) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
32	2, 30, 31	sylancr 586	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
33		fnfvima 7270	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
34	21, 33	mp3an1 1448	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
35	34	adantlr 714	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
36		onnmin 7834	. . . . . . . 8 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday ‘𝑞) ∈ ( bday “ 𝑎)) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
37	5, 35, 36	sylancr 586	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
38	37	ralrimiva 3152	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
39		eleq2 2833	. . . . . . . 8 ⊢ (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → (( bday ‘𝑞) ∈ ( bday ‘𝑝) ↔ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎)))
40	39	notbid 318	. . . . . . 7 ⊢ (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → (¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝) ↔ ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎)))
41	40	ralbidv 3184	. . . . . 6 ⊢ (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝) ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎)))
42	38, 41	syl5ibrcom 247	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
43	42	reximdv 3176	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
44	32, 43	mpd 15	. . 3 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝))
45		simpll 766	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No )
46		simprr 772	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 4009	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No )
48		simprl 770	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 4009	. . . . . . . 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 27991	. . . . . . . 8 ⊢ ((𝑞 ∈ No ∧ 𝑝 ∈ No ) → (𝑞𝑅𝑝 ↔ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
52	47, 49, 51	syl2anc 583	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → (𝑞𝑅𝑝 ↔ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
53	52	notbid 318	. . . . . 6 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
54	53	anassrs 467	. . . . 5 ⊢ ((((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) ∧ 𝑞 ∈ 𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
55	54	ralbidva 3182	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
56	55	rexbidva 3183	. . 3 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
57	44, 56	mpbird 257	. 2 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝)
58	1, 57	mpgbir 1797	1 ⊢ 𝑅 Fr No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  ∩ cint 4970   class class class wbr 5166  {copab 5228   Fr wfr 5649  ran crn 5701   “ cima 5703  Oncon0 6395  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573   No csur 27702   bday cbday 27704   L cleft 27902   R cright 27903

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-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  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-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  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-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  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-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-made 27904  df-old 27905  df-left 27907  df-right 27908

This theorem is referenced by:  noinds  27996  norecfn  27997  norecov  27998  noxpordfr  28002  no2indslem  28005  no3inds  28009