Theorem elovmpowrd 14594

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 𝑣 ∣ 𝜑})

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.

Step	Hyp	Ref	Expression
1		elovmpowrd.o	. . . 4 ⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑})
2		csbwrdg 14580	. . . . . . . 8 ⊢ (𝑣 ∈ V → ⦋𝑣 / 𝑥⦌Word 𝑥 = Word 𝑣)
3	2	eqcomd 2775	. . . . . . 7 ⊢ (𝑣 ∈ V → Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥)
4	3	adantr 485	. . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑦 ∈ V) → Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥)
5	4	rabeqdv 3438	. . . . 5 ⊢ ((𝑣 ∈ V ∧ 𝑦 ∈ V) → {𝑧 ∈ Word 𝑣 ∣ 𝜑} = {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
6	5	mpoeq3ia 7489	. . . 4 ⊢ (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑}) = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
7	1, 6	eqtri 2792	. . 3 ⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
8		csbwrdg 14580	. . . . 5 ⊢ (𝑉 ∈ V → ⦋𝑉 / 𝑥⦌Word 𝑥 = Word 𝑉)
9		wrdexg 14560	. . . . 5 ⊢ (𝑉 ∈ V → Word 𝑉 ∈ V)
10	8, 9	eqeltrd 2869	. . . 4 ⊢ (𝑉 ∈ V → ⦋𝑉 / 𝑥⦌Word 𝑥 ∈ V)
11	10	adantr 485	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑉 / 𝑥⦌Word 𝑥 ∈ V)
12	7, 11	elovmporab1w 7658	. 2 ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥))
13	8	eleq2d 2855	. . . . 5 ⊢ (𝑉 ∈ V → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 ↔ 𝑍 ∈ Word 𝑉))
14	13	adantr 485	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 ↔ 𝑍 ∈ Word 𝑉))
15		id 23	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
16	15	3expia 1137	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
17	14, 16	sylbid 243	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
18	17	3impia 1133	. 2 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
19	12, 18	syl 18	1 ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463  ⦋csb 3861  (class class class)co 7411   ∈ cmpo 7413  Word cword 14549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-1cn 11157  ax-addcl 11159

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-map 8825  df-nn 12233  df-n0 12504  df-word 14550

This theorem is referenced by: (None)