Theorem lrrecfr 28147

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 5613	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 27951	. . . . 5 ⊢ Fun bday
3		imassrn 6072	. . . . . . 7 ⊢ ( bday “ 𝑎) ⊆ ran bday
4		bdayrn 27955	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 3984	. . . . . 6 ⊢ ( bday “ 𝑎) ⊆ On
6		fvex 6894	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑞) ∈ V
7	6	jctr 533	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
8	7	eximi 1864	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ ( bday ‘𝑞) ∈ V))
9		n0 4306	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3089	. . . . . . . . . . 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 3468	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑞) ∈ V ↔ ∃𝑝 𝑝 = ( bday ‘𝑞))
13		eqcom 2769	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ( bday ‘𝑞) ↔ ( bday ‘𝑞) = 𝑝)
14	13	exbii 1877	. . . . . . . . . . . . 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 3111	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 ( bday ‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝( bday ‘𝑞) = 𝑝)
17		rexcom4 3291	. . . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝)
21		bdayfn 27952	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6953	. . . . . . . . . . 11 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
23	21, 22	mpan 702	. . . . . . . . . 10 ⊢ (𝑎 ⊆ No → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
24	23	adantr 485	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (𝑝 ∈ ( bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 ( bday ‘𝑞) = 𝑝))
25	24	exbidv 1950	. . . . . . . 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 4306	. . . . . . 7 ⊢ (( bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ ( bday “ 𝑎))
28	26, 27	sylibr 237	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ( bday “ 𝑎) ≠ ∅)
29		onint 7787	. . . . . 6 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday “ 𝑎) ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
30	5, 28, 29	sylancr 598	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎))
31		fvelima 6946	. . . . 5 ⊢ ((Fun bday ∧ ∩ ( bday “ 𝑎) ∈ ( bday “ 𝑎)) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
32	2, 30, 31	sylancr 598	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ( bday ‘𝑝) = ∩ ( bday “ 𝑎))
33		fnfvima 7231	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
34	21, 33	mp3an1 1476	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
35	34	adantlr 727	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ( bday ‘𝑞) ∈ ( bday “ 𝑎))
36		onnmin 7795	. . . . . . . 8 ⊢ ((( bday “ 𝑎) ⊆ On ∧ ( bday ‘𝑞) ∈ ( bday “ 𝑎)) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
37	5, 35, 36	sylancr 598	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
38	37	ralrimiva 3156	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ∩ ( bday “ 𝑎))
39		eleq2 2851	. . . . . . . 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 3187	. . . . . 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 3179	. . . 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 778	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No )
46		simprr 784	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 3937	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No )
48		simprl 782	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 3937	. . . . . . . 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 28144	. . . . . . . 8 ⊢ ((𝑞 ∈ No ∧ 𝑝 ∈ No ) → (𝑞𝑅𝑝 ↔ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
52	47, 49, 51	syl2anc 595	. . . . . . 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 472	. . . . 5 ⊢ ((((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) ∧ 𝑞 ∈ 𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
55	54	ralbidva 3185	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ ( bday ‘𝑞) ∈ ( bday ‘𝑝)))
56	55	rexbidva 3186	. . 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 1828	1 ⊢ 𝑅 Fr No

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  ∩ cint 4911   class class class wbr 5108  {copab 5172   Fr wfr 5610  ran crn 5661   “ cima 5663  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536   No csur 27815   bday cbday 27817   L cleft 28029   R cright 28030

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-1o 8451  df-2o 8452  df-no 27818  df-lts 27819  df-bday 27820  df-slts 27962  df-cuts 27964  df-made 28031  df-old 28032  df-left 28034  df-right 28035

This theorem is used by:  noinds  28149  norecfn  28150  norecov  28151  noxpordfr  28155  no2indlesm  28158  no3inds  28162