Theorem opfi1uzind 14560

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 14561) 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 4899	. . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝑉, 𝐸⟩)
4	3	eleq1d 2829	. . . . . 6 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
5	2, 4	sbcied 3850	. . . . 5 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
6	5	sbcieg 3845	. . . 4 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
7	6	biimparc 479	. . 3 ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺)
8	7	3adant3 1132	. 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 3492	. . . . . 6 ⊢ 𝑣 ∈ V
14		vex 3492	. . . . . 6 ⊢ 𝑒 ∈ V
15		opeq12 4899	. . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → ⟨𝑎, 𝑏⟩ = ⟨𝑣, 𝑒⟩)
16	15	eleq1d 2829	. . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺))
17	13, 14, 16	sbc2ie 3887	. . . . 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 580	. . . 4 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
20	13	difexi 5348	. . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V
21		opeq12 4899	. . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨(𝑣 ∖ {𝑛}), 𝐹⟩)
22	21	eleq1d 2829	. . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺))
23	20, 9, 22	sbc2ie 3887	. . . 4 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
24	19, 23	sylibr 234	. . 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 580	. . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
28	17	3anbi1i 1157	. . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
29	28	anbi2i 622	. . . 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 580	. . 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 14556	. 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]⟨𝑎, 𝑏⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
33	8, 32	syld3an1 1410	1 ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  [wsbc 3804   ∖ cdif 3973  {csn 4648  ⟨cop 4654   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  1c1 11185   + caddc 11187   ≤ cle 11325  ℕ₀cn0 12553  ♯chash 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-hash 14380

This theorem is referenced by:  opfi1ind  14561