Theorem lrrecfr 27878

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 5635	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 27723	. . . . 5 ⊢ Fun bday
3		imassrn 6077	. . . . . . 7 ⊢ ( bday “ 𝑎) ⊆ ran bday
4		bdayrn 27726	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 4016	. . . . . 6 ⊢ ( bday “ 𝑎) ⊆ On
6		fvex 6913	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑞) ∈ V
7	6	jctr 523	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
8	7	eximi 1829	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
9		n0 4348	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3067	. . . . . . . . . . 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 3484	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday ‘𝑞))
13		eqcom 2734	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ( bday ‘𝑞) ↔ ( bday ‘𝑞) = 𝑝)
14	13	exbii 1842	. . . . . . . . . . . . 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 3090	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝)
17		rexcom4 3281	. . . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
21		bdayfn 27724	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6974	. . . . . . . . . . 11 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
23	21, 22	mpan 688	. . . . . . . . . 10 ⊢ (𝑎 ⊆ No → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
24	23	adantr 479	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
25	24	exbidv 1916	. . . . . . . 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 4348	. . . . . . 7 ⊢ (( bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
28	26, 27	sylibr 233	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ( bday “ 𝑎) ≠ ∅)
29		onint 7797	. . . . . 6 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday “ 𝑎) ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
30	5, 28, 29	sylancr 585	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
31		fvelima 6967	. . . . 5 ⊢ ((Fun bday ∧ ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎)) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
32	2, 30, 31	sylancr 585	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
33		fnfvima 7249	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
34	21, 33	mp3an1 1444	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
35	34	adantlr 713	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
36		onnmin 7805	. . . . . . . 8 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday ‘𝑞) ∈ ( bday “ 𝑎)) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
37	5, 35, 36	sylancr 585	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
38	37	ralrimiva 3142	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
39		eleq2 2817	. . . . . . . 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 3173	. . . . . 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 3166	. . . 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 765	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No )
46		simprr 771	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 3981	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No )
48		simprl 769	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 3981	. . . . . . . 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 27875	. . . . . . . 8 ⊢ ((𝑞 ∈ No ∧ 𝑝 ∈ No ) → (𝑞𝑅𝑝 ↔ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
52	47, 49, 51	syl2anc 582	. . . . . . 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 466	. . . . 5 ⊢ ((((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) ∧ 𝑞 ∈ 𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
55	54	ralbidva 3171	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
56	55	rexbidva 3172	. . 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 1793	1 ⊢ 𝑅 Fr No

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2936  ∀wral 3057  ∃wrex 3066  Vcvv 3471   ∪ cun 3945   ⊆ wss 3947  ∅c0 4324  ∩ cint 4951   class class class wbr 5150  {copab 5212   Fr wfr 5632  ran crn 5681   “ cima 5683  Oncon0 6372  Fun wfun 6545   Fn wfn 6546  ‘cfv 6551   No csur 27591   bday cbday 27593   L cleft 27790   R cright 27791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-1o 8491  df-2o 8492  df-no 27594  df-slt 27595  df-bday 27596  df-sslt 27732  df-scut 27734  df-made 27792  df-old 27793  df-left 27795  df-right 27796

This theorem is referenced by:  noinds  27880  norecfn  27881  norecov  27882  noxpordfr  27886  no2indslem  27889  no3inds  27893