Theorem lrrecfr 34027

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 5535	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 33894	. . . . 5 ⊢ Fun bday
3		imassrn 5969	. . . . . . 7 ⊢ ( bday “ 𝑎) ⊆ ran bday
4		bdayrn 33897	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 3953	. . . . . 6 ⊢ ( bday “ 𝑎) ⊆ On
6		fvex 6769	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑞) ∈ V
7	6	jctr 524	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
8	7	eximi 1838	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
9		n0 4277	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3069	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
11	8, 9, 10	3imtr4i 291	. . . . . . . . . 10 ⊢ (𝑎 ≠ ∅ → ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V)
12		isset 3435	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday ‘𝑞))
13		eqcom 2745	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ( bday ‘𝑞) ↔ ( bday ‘𝑞) = 𝑝)
14	13	exbii 1851	. . . . . . . . . . . . 13 ⊢ (∃𝑝 𝑝 = ( bday ‘𝑞) ↔ ∃𝑝( bday ‘𝑞) = 𝑝)
15	12, 14	bitri 274	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝( bday ‘𝑞) = 𝑝)
16	15	rexbii 3177	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝)
17		rexcom4 3179	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝 ↔ ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
18	16, 17	bitri 274	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
19	11, 18	sylib 217	. . . . . . . . 9 ⊢ (𝑎 ≠ ∅ → ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
21		bdayfn 33895	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6823	. . . . . . . . . . 11 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
23	21, 22	mpan 686	. . . . . . . . . 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 1925	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
26	20, 25	mpbird 256	. . . . . . 7 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
27		n0 4277	. . . . . . 7 ⊢ (( bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
28	26, 27	sylibr 233	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ( bday “ 𝑎) ≠ ∅)
29		onint 7617	. . . . . 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 6817	. . . . 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 7091	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
34	21, 33	mp3an1 1446	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
35	34	adantlr 711	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
36		onnmin 7625	. . . . . . . 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 3107	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
39		eleq2 2827	. . . . . . . 8 ⊢ (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → (( bday ‘𝑞) ∈ ( bday ‘𝑝) ↔ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎)))
40	39	notbid 317	. . . . . . 7 ⊢ (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → (¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝) ↔ ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎)))
41	40	ralbidv 3120	. . . . . 6 ⊢ (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝) ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎)))
42	38, 41	syl5ibrcom 246	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (( bday ‘𝑝) = ∩ ( bday “ 𝑎) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
43	42	reximdv 3201	. . . 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 763	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No )
46		simprr 769	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 3918	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No )
48		simprl 767	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 3918	. . . . . . . 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 34024	. . . . . . . 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 317	. . . . . 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 3119	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
56	55	rexbidva 3224	. . 3 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
57	44, 56	mpbird 256	. 2 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝)
58	1, 57	mpgbir 1803	1 ⊢ 𝑅 Fr No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  ∩ cint 4876   class class class wbr 5070  {copab 5132   Fr wfr 5532  ran crn 5581   “ cima 5583  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418   No csur 33770   bday cbday 33772   L cleft 33956   R cright 33957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959  df-left 33961  df-right 33962

This theorem is referenced by:  noinds  34029  norecfn  34030  norecov  34031  noxpordfr  34035  no2indslem  34038  no3inds  34042