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Theorem elovmpt2wrd 13202
 Description: Implications for the value of an operation defined by the maps-to notation with a class abstration 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
elovmpt2wrd.o 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑})
Assertion
Ref Expression
elovmpt2wrd (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 elovmpt2wrd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elovmpt2wrd.o . . . 4 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑})
2 csbwrdg 13189 . . . . . . . 8 (𝑣 ∈ V → 𝑣 / 𝑥Word 𝑥 = Word 𝑣)
32eqcomd 2616 . . . . . . 7 (𝑣 ∈ V → Word 𝑣 = 𝑣 / 𝑥Word 𝑥)
43adantr 480 . . . . . 6 ((𝑣 ∈ V ∧ 𝑦 ∈ V) → Word 𝑣 = 𝑣 / 𝑥Word 𝑥)
5 rabeq 3166 . . . . . 6 (Word 𝑣 = 𝑣 / 𝑥Word 𝑥 → {𝑧 ∈ Word 𝑣𝜑} = {𝑧𝑣 / 𝑥Word 𝑥𝜑})
64, 5syl 17 . . . . 5 ((𝑣 ∈ V ∧ 𝑦 ∈ V) → {𝑧 ∈ Word 𝑣𝜑} = {𝑧𝑣 / 𝑥Word 𝑥𝜑})
76mpt2eq3ia 6618 . . . 4 (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑}) = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑣 / 𝑥Word 𝑥𝜑})
81, 7eqtri 2632 . . 3 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑣 / 𝑥Word 𝑥𝜑})
9 csbwrdg 13189 . . . . 5 (𝑉 ∈ V → 𝑉 / 𝑥Word 𝑥 = Word 𝑉)
10 wrdexg 13170 . . . . 5 (𝑉 ∈ V → Word 𝑉 ∈ V)
119, 10eqeltrd 2688 . . . 4 (𝑉 ∈ V → 𝑉 / 𝑥Word 𝑥 ∈ V)
1211adantr 480 . . 3 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → 𝑉 / 𝑥Word 𝑥 ∈ V)
138, 12elovmpt2rab1 6779 . 2 (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑉 / 𝑥Word 𝑥))
149eleq2d 2673 . . . . 5 (𝑉 ∈ V → (𝑍𝑉 / 𝑥Word 𝑥𝑍 ∈ Word 𝑉))
1514adantr 480 . . . 4 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍𝑉 / 𝑥Word 𝑥𝑍 ∈ Word 𝑉))
16 id 22 . . . . 5 ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
17163expia 1259 . . . 4 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
1815, 17sylbid 229 . . 3 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍𝑉 / 𝑥Word 𝑥 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
19183impia 1253 . 2 ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑉 / 𝑥Word 𝑥) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
2013, 19syl 17 1 (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⦋csb 3499  (class class class)co 6549   ↦ cmpt2 6551  Word cword 13146 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-pm 7747  df-neg 10148  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-word 13154 This theorem is referenced by:  wwlkprop  26213
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