Theorem opfi1uzind 14657

Description: Properties of an ordered pair 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 (represented as ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14658) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

Hypotheses

Ref	Expression
opfi1uzind.e	⊢ 𝐸 ∈ V
opfi1uzind.f	⊢ 𝐹 ∈ V
opfi1uzind.l	⊢ 𝐿 ∈ ℕ0
opfi1uzind.1	⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
opfi1uzind.2	⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
opfi1uzind.3	⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
opfi1uzind.4	⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
opfi1uzind.base	⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
opfi1uzind.step	⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)

Assertion

Ref	Expression
opfi1uzind	⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

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

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 opfi1uzind

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

Step	Hyp	Ref	Expression
1		opfi1uzind.e	. . . . . . 7 ⊢ 𝐸 ∈ V
2	1	a1i 11	. . . . . 6 ⊢ (𝑎 = 𝑉 → 𝐸 ∈ V)
3		opeq12 4834	. . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝑉, 𝐸⟩)
4	3	eleq1d 2845	. . . . . 6 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
5	2, 4	sbcied 3781	. . . . 5 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
6	5	sbcieg 3777	. . . 4 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
7	6	biimparc 485	. . 3 ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺)
8	7	3adant3 1150	. 2 ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → [𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺)
9		opfi1uzind.f	. . 3 ⊢ 𝐹 ∈ V
10		opfi1uzind.l	. . 3 ⊢ 𝐿 ∈ ℕ0
11		opfi1uzind.1	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
12		opfi1uzind.2	. . 3 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
13		vex 3454	. . . . . 6 ⊢ 𝑣 ∈ V
14		vex 3454	. . . . . 6 ⊢ 𝑒 ∈ V
15		opeq12 4834	. . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → ⟨𝑎, 𝑏⟩ = ⟨𝑣, 𝑒⟩)
16	15	eleq1d 2845	. . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺))
17	13, 14, 16	sbc2ie 3813	. . . . 5 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺)
18		opfi1uzind.3	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
19	17, 18	sylanb 593	. . . 4 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
20	13	difexi 5291	. . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V
21		opeq12 4834	. . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨(𝑣 ∖ {𝑛}), 𝐹⟩)
22	21	eleq1d 2845	. . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺))
23	20, 9, 22	sbc2ie 3813	. . . 4 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
24	19, 23	sylibr 237	. . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺)
25		opfi1uzind.4	. . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
26		opfi1uzind.base	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
27	17, 26	sylanb 593	. . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
28	17	3anbi1i 1175	. . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
29	28	anbi2i 635	. . . 4 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)))
30		opfi1uzind.step	. . . 4 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
31	29, 30	sylanb 593	. . 3 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
32	9, 10, 11, 12, 24, 25, 27, 31	fi1uzind 14653	. 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
33	8, 32	syld3an1 1437	1 ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450  [wsbc 3738   ∖ cdif 4045  {csn 4583  ⟨cop 4589   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Fincfn 8974  1c1 11202   + caddc 11204   ≤ cle 11345  ℕ₀cn0 12607  ♯chash 14475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-oadd 8481  df-er 8718  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-dju 9983  df-card 10021  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-n0 12608  df-xnn0 12681  df-z 12695  df-uz 12967  df-fz 13641  df-hash 14476

This theorem is used by:  opfi1ind  14658