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Theorem elovmpowrd 14143
Description: Implications for the value of an operation defined by the maps-to notation with a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypothesis
Ref Expression
elovmpowrd.o 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑})
Assertion
Ref Expression
elovmpowrd (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
Distinct variable groups:   𝑣,𝑉,𝑦,𝑧   𝑣,𝑌,𝑦,𝑧   𝑧,𝑍
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣)   𝑂(𝑦,𝑧,𝑣)   𝑍(𝑦,𝑣)

Proof of Theorem elovmpowrd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elovmpowrd.o . . . 4 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑})
2 csbwrdg 14129 . . . . . . . 8 (𝑣 ∈ V → 𝑣 / 𝑥Word 𝑥 = Word 𝑣)
32eqcomd 2745 . . . . . . 7 (𝑣 ∈ V → Word 𝑣 = 𝑣 / 𝑥Word 𝑥)
43adantr 484 . . . . . 6 ((𝑣 ∈ V ∧ 𝑦 ∈ V) → Word 𝑣 = 𝑣 / 𝑥Word 𝑥)
54rabeqdv 3409 . . . . 5 ((𝑣 ∈ V ∧ 𝑦 ∈ V) → {𝑧 ∈ Word 𝑣𝜑} = {𝑧𝑣 / 𝑥Word 𝑥𝜑})
65mpoeq3ia 7310 . . . 4 (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑}) = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑣 / 𝑥Word 𝑥𝜑})
71, 6eqtri 2767 . . 3 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑣 / 𝑥Word 𝑥𝜑})
8 csbwrdg 14129 . . . . 5 (𝑉 ∈ V → 𝑉 / 𝑥Word 𝑥 = Word 𝑉)
9 wrdexg 14109 . . . . 5 (𝑉 ∈ V → Word 𝑉 ∈ V)
108, 9eqeltrd 2840 . . . 4 (𝑉 ∈ V → 𝑉 / 𝑥Word 𝑥 ∈ V)
1110adantr 484 . . 3 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → 𝑉 / 𝑥Word 𝑥 ∈ V)
127, 11elovmporab1w 7473 . 2 (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑉 / 𝑥Word 𝑥))
138eleq2d 2825 . . . . 5 (𝑉 ∈ V → (𝑍𝑉 / 𝑥Word 𝑥𝑍 ∈ Word 𝑉))
1413adantr 484 . . . 4 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍𝑉 / 𝑥Word 𝑥𝑍 ∈ Word 𝑉))
15 id 22 . . . . 5 ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
16153expia 1123 . . . 4 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
1714, 16sylbid 243 . . 3 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍𝑉 / 𝑥Word 𝑥 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
18173impia 1119 . 2 ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑉 / 𝑥Word 𝑥) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
1912, 18syl 17 1 (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2112  {crab 3067  Vcvv 3422  csb 3827  (class class class)co 7234  cmpo 7236  Word cword 14099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544  ax-cnex 10812  ax-1cn 10814  ax-addcl 10816
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3711  df-csb 3828  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-om 7666  df-wrecs 8070  df-recs 8131  df-rdg 8169  df-map 8533  df-nn 11858  df-n0 12118  df-word 14100
This theorem is referenced by: (None)
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