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Theorem opfi1uzindOLD 13144
Description: Obsolete version of opfi1uzind 13138 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 4342 . . . . . . . 8 ((𝑎 = 𝑉𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝑉, 𝐸⟩)
54eleq1d 2672 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
63, 5sbcied 3439 . . . . . 6 (𝑎 = 𝑉 → ([𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
76adantl 481 . . . . 5 ((𝑉 ∈ Fin ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
81, 7sbcied 3439 . . . 4 (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
98biimparc 503 . . 3 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
1093adant3 1074 . 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 3176 . . . . . 6 𝑣 ∈ V
16 vex 3176 . . . . . 6 𝑒 ∈ V
17 opeq12 4342 . . . . . . 7 ((𝑎 = 𝑣𝑏 = 𝑒) → ⟨𝑎, 𝑏⟩ = ⟨𝑣, 𝑒⟩)
1817eleq1d 2672 . . . . . 6 ((𝑎 = 𝑣𝑏 = 𝑒) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺))
1915, 16, 18sbc2ie 3472 . . . . 5 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺)
20 opfi1uzindOLD.3 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
2119, 20sylanb 488 . . . 4 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
22 difexg 4735 . . . . . 6 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
2315, 22ax-mp 5 . . . . 5 (𝑣 ∖ {𝑛}) ∈ V
2411elexi 3186 . . . . 5 𝐹 ∈ V
25 opeq12 4342 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨(𝑣 ∖ {𝑛}), 𝐹⟩)
2625eleq1d 2672 . . . . 5 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺))
2723, 24, 26sbc2ie 3472 . . . 4 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
2821, 27sylibr 223 . . 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 488 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)
33193anbi1i 1246 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
3433anbi2i 726 . . . 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 488 . . 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 13140 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
3810, 37syld3an1 1364 1 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  [wsbc 3402  cdif 3537  {csn 4125  cop 4131   class class class wbr 4583  cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   + caddc 9818  cle 9954  0cn0 11169  #chash 12979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980
This theorem is referenced by:  opfi1indOLD  13145
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