Theorem brfi1uzind 14550

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 14551) 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 5716	. . 3 ⊢ (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3		simpl 487	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
4		simplr 780	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → 𝐸 ∈ V)
5		breq12 5114	. . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
6	5	adantll 726	. . . . . 6 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
7	4, 6	sbcied 3787	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎𝐺𝑏 ↔ 𝑉𝐺𝐸))
8	3, 7	sbcied 3787	. . . 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 3459	. . . . 5 ⊢ 𝑣 ∈ V
16		vex 3459	. . . . 5 ⊢ 𝑒 ∈ V
17		breq12 5114	. . . . 5 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (𝑎𝐺𝑏 ↔ 𝑣𝐺𝑒))
18	15, 16, 17	sbc2ie 3819	. . . 4 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ↔ 𝑣𝐺𝑒)
19		brfi1uzind.3	. . . . 5 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
20	15	difexi 5301	. . . . . 6 ⊢ (𝑣 ∖ {𝑛}) ∈ V
21		breq12 5114	. . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
22	20, 11, 21	sbc2ie 3819	. . . . 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 592	. . 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 592	. . 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 634	. . . 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 592	. . 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 14549	. 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 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455  [wsbc 3744   ∖ cdif 3902  {csn 4589   class class class wbr 5109  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  1c1 11105   + caddc 11107   ≤ cle 11248  ℕ₀cn0 12508  ♯chash 14371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9892  df-card 9930  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-n0 12509  df-xnn0 12582  df-z 12596  df-uz 12867  df-fz 13540  df-hash 14372

This theorem is used by:  brfi1ind  14551