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Theorem opfi1uzindOLD 13223
Description: Obsolete version of opfi1uzind 13217 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
opfi1uzindOLD.e 𝐸𝑌
opfi1uzindOLD.f 𝐹𝑈
opfi1uzindOLD.l 𝐿 ∈ ℕ0
opfi1uzindOLD.1 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
opfi1uzindOLD.2 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
opfi1uzindOLD.3 ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
opfi1uzindOLD.4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
opfi1uzindOLD.base ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)
opfi1uzindOLD.step ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
Assertion
Ref Expression
opfi1uzindOLD ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ 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)   𝑈(𝑦,𝑤,𝑣,𝑒,𝑓,𝑛)   𝐸(𝑦,𝑤,𝑓)   𝐹(𝑦,𝑣,𝑒,𝑛)   𝐿(𝑤,𝑓)   𝑉(𝑦,𝑤,𝑓)   𝑌(𝑦,𝑤,𝑣,𝑒,𝑓,𝑛)

Proof of Theorem opfi1uzindOLD
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑉 ∈ Fin → 𝑉 ∈ Fin)
2 opfi1uzindOLD.e . . . . . . . 8 𝐸𝑌
32a1i 11 . . . . . . 7 (𝑎 = 𝑉𝐸𝑌)
4 opeq12 4377 . . . . . . . 8 ((𝑎 = 𝑉𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝑉, 𝐸⟩)
54eleq1d 2688 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
63, 5sbcied 3459 . . . . . 6 (𝑎 = 𝑉 → ([𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
76adantl 482 . . . . 5 ((𝑉 ∈ Fin ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
81, 7sbcied 3459 . . . 4 (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
98biimparc 504 . . 3 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
1093adant3 1079 . 2 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
11 opfi1uzindOLD.f . . 3 𝐹𝑈
12 opfi1uzindOLD.l . . 3 𝐿 ∈ ℕ0
13 opfi1uzindOLD.1 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
14 opfi1uzindOLD.2 . . 3 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
15 vex 3194 . . . . . 6 𝑣 ∈ V
16 vex 3194 . . . . . 6 𝑒 ∈ V
17 opeq12 4377 . . . . . . 7 ((𝑎 = 𝑣𝑏 = 𝑒) → ⟨𝑎, 𝑏⟩ = ⟨𝑣, 𝑒⟩)
1817eleq1d 2688 . . . . . 6 ((𝑎 = 𝑣𝑏 = 𝑒) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺))
1915, 16, 18sbc2ie 3492 . . . . 5 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺)
20 opfi1uzindOLD.3 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
2119, 20sylanb 489 . . . 4 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
22 difexg 4773 . . . . . 6 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
2315, 22ax-mp 5 . . . . 5 (𝑣 ∖ {𝑛}) ∈ V
2411elexi 3204 . . . . 5 𝐹 ∈ V
25 opeq12 4377 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨(𝑣 ∖ {𝑛}), 𝐹⟩)
2625eleq1d 2688 . . . . 5 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺))
2723, 24, 26sbc2ie 3492 . . . 4 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
2821, 27sylibr 224 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
29 opfi1uzindOLD.4 . . . 4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
3029idi 2 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
31 opfi1uzindOLD.base . . . 4 ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)
3219, 31sylanb 489 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)
33193anbi1i 1251 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
3433anbi2i 729 . . . 4 (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
35 opfi1uzindOLD.step . . . 4 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3634, 35sylanb 489 . . 3 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3711, 12, 13, 14, 28, 30, 32, 36fi1uzindOLD 13219 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
3810, 37syld3an1 1369 1 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  [wsbc 3422  cdif 3557  {csn 4153  cop 4159   class class class wbr 4618  cfv 5850  (class class class)co 6605  Fincfn 7900  1c1 9882   + caddc 9884  cle 10020  0cn0 11237  #chash 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055
This theorem is referenced by:  opfi1indOLD  13224
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