Theorem elovmpt2wrd 13347

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 𝑣 ∣ 𝜑})

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.

Step	Hyp	Ref	Expression
1		elovmpt2wrd.o	. . . 4 ⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑})
2		csbwrdg 13334	. . . . . . . 8 ⊢ (𝑣 ∈ V → ⦋𝑣 / 𝑥⦌Word 𝑥 = Word 𝑣)
3	2	eqcomd 2628	. . . . . . 7 ⊢ (𝑣 ∈ V → Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥)
4	3	adantr 481	. . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑦 ∈ V) → Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥)
5		rabeq 3192	. . . . . 6 ⊢ (Word 𝑣 = ⦋𝑣 / 𝑥⦌Word 𝑥 → {𝑧 ∈ Word 𝑣 ∣ 𝜑} = {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
6	4, 5	syl 17	. . . . 5 ⊢ ((𝑣 ∈ V ∧ 𝑦 ∈ V) → {𝑧 ∈ Word 𝑣 ∣ 𝜑} = {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
7	6	mpt2eq3ia 6720	. . . 4 ⊢ (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑}) = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
8	1, 7	eqtri 2644	. . 3 ⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑣 / 𝑥⦌Word 𝑥 ∣ 𝜑})
9		csbwrdg 13334	. . . . 5 ⊢ (𝑉 ∈ V → ⦋𝑉 / 𝑥⦌Word 𝑥 = Word 𝑉)
10		wrdexg 13315	. . . . 5 ⊢ (𝑉 ∈ V → Word 𝑉 ∈ V)
11	9, 10	eqeltrd 2701	. . . 4 ⊢ (𝑉 ∈ V → ⦋𝑉 / 𝑥⦌Word 𝑥 ∈ V)
12	11	adantr 481	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑉 / 𝑥⦌Word 𝑥 ∈ V)
13	8, 12	elovmpt2rab1 6881	. 2 ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥))
14	9	eleq2d 2687	. . . . 5 ⊢ (𝑉 ∈ V → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 ↔ 𝑍 ∈ Word 𝑉))
15	14	adantr 481	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 ↔ 𝑍 ∈ Word 𝑉))
16		id 22	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
17	16	3expia 1267	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
18	15, 17	sylbid 230	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥 → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉)))
19	18	3impia 1261	. 2 ⊢ ((𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑉 / 𝑥⦌Word 𝑥) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
20	13, 19	syl 17	1 ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  ⦋csb 3533  (class class class)co 6650   ↦ cmpt2 6652  Word cword 13291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-pm 7860  df-neg 10269  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-word 13299

This theorem is referenced by: (None)