Theorem lrrecfr 27935

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 5584	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 27740	. . . . 5 ⊢ Fun bday
3		imassrn 6036	. . . . . . 7 ⊢ ( bday “ 𝑎) ⊆ ran bday
4		bdayrn 27743	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 3970	. . . . . 6 ⊢ ( bday “ 𝑎) ⊆ On
6		fvex 6853	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑞) ∈ V
7	6	jctr 524	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
8	7	eximi 1837	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
9		n0 4293	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3062	. . . . . . . . . . 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 3443	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday ‘𝑞))
13		eqcom 2743	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ( bday ‘𝑞) ↔ ( bday ‘𝑞) = 𝑝)
14	13	exbii 1850	. . . . . . . . . . . . 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 3084	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝)
17		rexcom4 3264	. . . . . . . . . . 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 27741	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6912	. . . . . . . . . . 11 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
23	21, 22	mpan 691	. . . . . . . . . 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 1923	. . . . . . . 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 4293	. . . . . . 7 ⊢ (( bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
28	26, 27	sylibr 234	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ( bday “ 𝑎) ≠ ∅)
29		onint 7744	. . . . . 6 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday “ 𝑎) ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
30	5, 28, 29	sylancr 588	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
31		fvelima 6905	. . . . 5 ⊢ ((Fun bday ∧ ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎)) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
32	2, 30, 31	sylancr 588	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
33		fnfvima 7188	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
34	21, 33	mp3an1 1451	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
35	34	adantlr 716	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
36		onnmin 7752	. . . . . . . 8 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday ‘𝑞) ∈ ( bday “ 𝑎)) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
37	5, 35, 36	sylancr 588	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
38	37	ralrimiva 3129	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
39		eleq2 2825	. . . . . . . 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 3160	. . . . . 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 3152	. . . 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 767	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No )
46		simprr 773	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 3922	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No )
48		simprl 771	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 3922	. . . . . . . 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 27932	. . . . . . . 8 ⊢ ((𝑞 ∈ No ∧ 𝑝 ∈ No ) → (𝑞𝑅𝑝 ↔ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
52	47, 49, 51	syl2anc 585	. . . . . . 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 3158	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
56	55	rexbidva 3159	. . 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 1801	1 ⊢ 𝑅 Fr No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  ∩ cint 4889   class class class wbr 5085  {copab 5147   Fr wfr 5581  ran crn 5632   “ cima 5634  Oncon0 6323  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498   No csur 27603   bday cbday 27605   L cleft 27817   R cright 27818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823

This theorem is referenced by:  noinds  27937  norecfn  27938  norecov  27939  noxpordfr  27943  no2indlesm  27946  no3inds  27950