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Theorem elovmpt2wrd 13342
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 13329 . . . . . . . 8 (𝑣 ∈ V → 𝑣 / 𝑥Word 𝑥 = Word 𝑣)
32eqcomd 2627 . . . . . . 7 (𝑣 ∈ V → Word 𝑣 = 𝑣 / 𝑥Word 𝑥)
43adantr 481 . . . . . 6 ((𝑣 ∈ V ∧ 𝑦 ∈ V) → Word 𝑣 = 𝑣 / 𝑥Word 𝑥)
5 rabeq 3190 . . . . . 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 6717 . . . 4 (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑}) = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑣 / 𝑥Word 𝑥𝜑})
81, 7eqtri 2643 . . 3 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑣 / 𝑥Word 𝑥𝜑})
9 csbwrdg 13329 . . . . 5 (𝑉 ∈ V → 𝑉 / 𝑥Word 𝑥 = Word 𝑉)
10 wrdexg 13310 . . . . 5 (𝑉 ∈ V → Word 𝑉 ∈ V)
119, 10eqeltrd 2700 . . . 4 (𝑉 ∈ V → 𝑉 / 𝑥Word 𝑥 ∈ V)
1211adantr 481 . . 3 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → 𝑉 / 𝑥Word 𝑥 ∈ V)
138, 12elovmpt2rab1 6878 . 2 (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑉 / 𝑥Word 𝑥))
149eleq2d 2686 . . . . 5 (𝑉 ∈ V → (𝑍𝑉 / 𝑥Word 𝑥𝑍 ∈ Word 𝑉))
1514adantr 481 . . . 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 1266 . . . 4 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
1815, 17sylbid 230 . . 3 ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍𝑉 / 𝑥Word 𝑥 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
19183impia 1260 . 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 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  {crab 2915  Vcvv 3198  csb 3531  (class class class)co 6647  cmpt2 6649  Word cword 13286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856  df-pm 7857  df-neg 10266  df-z 11375  df-uz 11685  df-fz 12324  df-fzo 12462  df-word 13294
This theorem is referenced by: (None)
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