Theorem lrrecfr 28356

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 5608	. 2 ⊢ (𝑅 Fr No ↔ ∀𝑎((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 ∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝))
2		bdayfun 28160	. . . . 5 ⊢ Fun bday
3		imassrn 6209	. . . . . . 7 ⊢ (bday “ 𝑎) ⊆ ran bday
4		bdayrn 28164	. . . . . . 7 ⊢ ran bday = On
5	3, 4	sseqtri 3961	. . . . . 6 ⊢ (bday “ 𝑎) ⊆ On
6		fvex 6918	. . . . . . . . . . . . 13 ⊢ (bday‘𝑞) ∈ V
7	6	jctr 534	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝑎 → (𝑞 ∈ 𝑎 ∧ (bday‘𝑞) ∈ V))
8	7	eximi 1868	. . . . . . . . . . 11 ⊢ (∃𝑞 𝑞 ∈ 𝑎 → ∃𝑞(𝑞 ∈ 𝑎 ∧ (bday‘𝑞) ∈ V))
9		n0 4303	. . . . . . . . . . 11 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑞 𝑞 ∈ 𝑎)
10		df-rex 3091	. . . . . . . . . . 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 2770	. . . . . . . . . . . . . 14 ⊢ (𝑝 = (bday‘𝑞) ↔ (bday‘𝑞) = 𝑝)
14	13	exbii 1881	. . . . . . . . . . . . 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 3113	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑎 (bday‘𝑞) ∈ V ↔ ∃𝑞 ∈ 𝑎 ∃𝑝(bday‘𝑞) = 𝑝)
17		rexcom4 3293	. . . . . . . . . . 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 487	. . . . . . . 8 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝)
21		bdayfn 28161	. . . . . . . . . . 11 ⊢ bday Fn No
22		fvelimab 6977	. . . . . . . . . . 11 ⊢ ((bday Fn No ∧ 𝑎 ⊆ No) → (𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
23	21, 22	mpan 703	. . . . . . . . . 10 ⊢ (𝑎 ⊆ No → (𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
24	23	adantr 486	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (𝑝 ∈ (bday “ 𝑎) ↔ ∃𝑞 ∈ 𝑎 (bday‘𝑞) = 𝑝))
25	24	exbidv 1956	. . . . . . . 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 4303	. . . . . . 7 ⊢ ((bday “ 𝑎) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (bday “ 𝑎))
28	26, 27	sylibr 237	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → (bday “ 𝑎) ≠ ∅)
29		onint 7825	. . . . . 6 ⊢ (((bday “ 𝑎) ⊆ On ∧ (bday “ 𝑎) ≠ ∅) → ∩ (bday “ 𝑎) ∈ (bday “ 𝑎))
30	5, 28, 29	sylancr 599	. . . . 5 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∩ (bday “ 𝑎) ∈ (bday “ 𝑎))
31		fvelima 6970	. . . . 5 ⊢ ((Fun bday ∧ ∩ (bday “ 𝑎) ∈ (bday “ 𝑎)) → ∃𝑝 ∈ 𝑎 (bday‘𝑝) = ∩ (bday “ 𝑎))
32	2, 30, 31	sylancr 599	. . . 4 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∃𝑝 ∈ 𝑎 (bday‘𝑝) = ∩ (bday “ 𝑎))
33		fnfvima 7259	. . . . . . . . . 10 ⊢ ((bday Fn No ∧ 𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → (bday‘𝑞) ∈ (bday “ 𝑎))
34	21, 33	mp3an1 1478	. . . . . . . . 9 ⊢ ((𝑎 ⊆ No ∧ 𝑞 ∈ 𝑎) → (bday‘𝑞) ∈ (bday “ 𝑎))
35	34	adantlr 728	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → (bday‘𝑞) ∈ (bday “ 𝑎))
36		onnmin 7833	. . . . . . . 8 ⊢ (((bday “ 𝑎) ⊆ On ∧ (bday‘𝑞) ∈ (bday “ 𝑎)) → ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎))
37	5, 35, 36	sylancr 599	. . . . . . 7 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑞 ∈ 𝑎) → ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎))
38	37	ralrimiva 3158	. . . . . 6 ⊢ ((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) → ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ ∩ (bday “ 𝑎))
39		eleq2 2853	. . . . . . . 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 3189	. . . . . 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 3181	. . . 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 779	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑎 ⊆ No)
46		simprr 785	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ 𝑎)
47	45, 46	sseldd 3914	. . . . . . . 8 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑞 ∈ No)
48		simprl 783	. . . . . . . . 9 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ (𝑝 ∈ 𝑎 ∧ 𝑞 ∈ 𝑎)) → 𝑝 ∈ 𝑎)
49	45, 48	sseldd 3914	. . . . . . . 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 28353	. . . . . . . 8 ⊢ ((𝑞 ∈ No ∧ 𝑝 ∈ No) → (𝑞𝑅𝑝 ↔ (bday‘𝑞) ∈ (bday‘𝑝)))
52	47, 49, 51	syl2anc 596	. . . . . . 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 473	. . . . 5 ⊢ ((((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) ∧ 𝑞 ∈ 𝑎) → (¬ 𝑞𝑅𝑝 ↔ ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
55	54	ralbidva 3187	. . . 4 ⊢ (((𝑎 ⊆ No ∧ 𝑎 ≠ ∅) ∧ 𝑝 ∈ 𝑎) → (∀𝑞 ∈ 𝑎 ¬ 𝑞𝑅𝑝 ↔ ∀𝑞 ∈ 𝑎 ¬ (bday‘𝑞) ∈ (bday‘𝑝)))
56	55	rexbidva 3188	. . 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 1832	1 ⊢ 𝑅 Fr No

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  ∩ cint 4910   class class class wbr 5106  {copab 5170   Fr wfr 5605  ran crn 5656   “ cima 5658  Oncon0 6377  Fun wfun 6549   Fn wfn 6550  ‘cfv 6555  Nocsur 28024  bdaycbday 28026  Lcleft 28238  Rcright 28239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244

This theorem is used by:  noinds  28358  norecfn  28359  norecov  28360  noxpordfr  28364  no2indlesm  28367  no3inds  28371