Theorem brfi1uzind 14598

Description: Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with 𝐿 = 0 (see brfi1ind 14599) or 𝐿 = 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Proof shortened by AV, 23-Oct-2020.) (Revised by AV, 28-Mar-2021.)

Hypotheses

Ref	Expression
brfi1uzind.r	⊢ Rel 𝐺
brfi1uzind.f	⊢ 𝐹 ∈ V
brfi1uzind.l	⊢ 𝐿 ∈ ℕ0
brfi1uzind.1	⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
brfi1uzind.2	⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
brfi1uzind.3	⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
brfi1uzind.4	⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
brfi1uzind.base	⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
brfi1uzind.step	⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)

Assertion

Ref	Expression
brfi1uzind	⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Distinct variable groups:   𝑒,𝐸,𝑛,𝑣   𝑓,𝐹,𝑤   𝑒,𝐺,𝑓,𝑛,𝑣,𝑤,𝑦   𝑒,𝐿,𝑛,𝑣,𝑦   𝑒,𝑉,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣

Allowed substitution hints:   𝜑(𝑦, 𝑤, 𝑓)   𝜓(𝑣, 𝑒)   𝜒(𝑦, 𝑣, 𝑒, 𝑛)   𝜃(𝑦, 𝑤, 𝑓)   𝐸(𝑦, 𝑤, 𝑓)   𝐹(𝑦, 𝑣, 𝑒, 𝑛)   𝐿(𝑤, 𝑓)   𝑉(𝑦, 𝑤, 𝑓)

Proof of Theorem brfi1uzind

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brfi1uzind.r	. . . 4 ⊢ Rel 𝐺
2	1	brrelex12i 5710	. . 3 ⊢ (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3		simpl 488	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
4		simplr 781	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → 𝐸 ∈ V)
5		breq12 5108	. . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
6	5	adantll 727	. . . . . 6 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
7	4, 6	sbcied 3782	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
8	3, 7	sbcied 3782	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
9	8	biimprcd 253	. . 3 ⊢ (𝑉𝐺𝐸 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏))
10	2, 9	mpd 16	. 2 ⊢ (𝑉𝐺𝐸 → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏)
11		brfi1uzind.f	. . 3 ⊢ 𝐹 ∈ V
12		brfi1uzind.l	. . 3 ⊢ 𝐿 ∈ ℕ0
13		brfi1uzind.1	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
14		brfi1uzind.2	. . 3 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
15		vex 3454	. . . . 5 ⊢ 𝑣 ∈ V
16		vex 3454	. . . . 5 ⊢ 𝑒 ∈ V
17		breq12 5108	. . . . 5 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (𝑎𝐺𝑏 ↔ 𝑣𝐺𝑒))
18	15, 16, 17	sbc2ie 3814	. . . 4 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ↔ 𝑣𝐺𝑒)
19		brfi1uzind.3	. . . . 5 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
20	15	difexi 5295	. . . . . 6 ⊢ (𝑣 ∖ {𝑛}) ∈ V
21		breq12 5108	. . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
22	20, 11, 21	sbc2ie 3814	. . . . 5 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹)
23	19, 22	sylibr 237	. . . 4 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
24	18, 23	sylanb 593	. . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
25		brfi1uzind.4	. . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
26		brfi1uzind.base	. . . 4 ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
27	18, 26	sylanb 593	. . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
28	18	3anbi1i 1175	. . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
29	28	anbi2i 635	. . . 4 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)))
30		brfi1uzind.step	. . . 4 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
31	29, 30	sylanb 593	. . 3 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
32	11, 12, 13, 14, 24, 25, 27, 31	fi1uzind 14597	. 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
33	10, 32	syl3an1 1181	1 ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450  [wsbc 3739   ∖ cdif 3896  {csn 4584   class class class wbr 5103  Rel wrel 5660  ‘cfv 6535  (class class class)co 7416  Fincfn 8959  1c1 11150   + caddc 11152   ≤ cle 11293  ℕ₀cn0 12553  ♯chash 14419

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-oadd 8466  df-er 8703  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9931  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-n0 12554  df-xnn0 12627  df-z 12641  df-uz 12913  df-fz 13587  df-hash 14420

This theorem is used by:  brfi1ind  14599