Theorem elovmpowrd 14523

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 14509	. . . . . . . 8 ⊢ (𝑣 ∈ V → ⦋𝑣 / 𝑥⦌Word 𝑥 = Word 𝑣)
3	2	eqcomd 2735	. . . . . . 7 ⊢ (𝑣 ∈ V → Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥)
4	3	adantr 480	. . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑦 ∈ V) → Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥)
5	4	rabeqdv 3421	. . . . 5 ⊢ ((𝑣 ∈ V ∧ 𝑦 ∈ V) → {𝑧 ∈ Word 𝑣 ∣ 𝜑} = {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
6	5	mpoeq3ia 7467	. . . 4 ⊢ (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑}) = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
7	1, 6	eqtri 2752	. . 3 ⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
8		csbwrdg 14509	. . . . 5 ⊢ (𝑉 ∈ V → ⦋𝑉 / 𝑥⦌Word 𝑥 = Word 𝑉)
9		wrdexg 14489	. . . . 5 ⊢ (𝑉 ∈ V → Word 𝑉 ∈ V)
10	8, 9	eqeltrd 2828	. . . 4 ⊢ (𝑉 ∈ V → ⦋𝑉 / 𝑥⦌Word 𝑥 ∈ V)
11	10	adantr 480	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑉 / 𝑥⦌Word 𝑥 ∈ V)
12	7, 11	elovmporab1w 7636	. 2 ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥))
13	8	eleq2d 2814	. . . . 5 ⊢ (𝑉 ∈ V → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 ↔ 𝑍 ∈ Word 𝑉))
14	13	adantr 480	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 ↔ 𝑍 ∈ Word 𝑉))
15		id 22	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
16	15	3expia 1121	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
17	14, 16	sylbid 240	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
18	17	3impia 1117	. 2 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
19	12, 18	syl 17	1 ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447  ⦋csb 3862  (class class class)co 7387   ∈ cmpo 7389  Word cword 14478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-map 8801  df-nn 12187  df-n0 12443  df-word 14479

This theorem is referenced by: (None)