mrsubvrs - Mathbox for Mario Carneiro

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Theorem mrsubvrs 32882

Description: The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mrsubco.s	⊢ 𝑆 = (mRSubst‘𝑇)
mrsubvrs.v	⊢ 𝑉 = (mVR‘𝑇)
mrsubvrs.r	⊢ 𝑅 = (mREx‘𝑇)

**Assertion**
Ref	Expression
mrsubvrs	⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅) → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))

Distinct variable groups: 𝑥,𝐹 𝑥,𝑆 𝑥,𝑇 𝑥,𝑉 𝑥,𝑋 Allowed substitution hint: 𝑅(𝑥)

Proof of Theorem mrsubvrs Dummy variables 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4249	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubco.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6639	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 143	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		eqid 2798	. . . . . 6 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
6		mrsubvrs.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
7		mrsubvrs.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
8	5, 6, 7	mrexval 32861	. . . . 5 ⊢ (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
9	4, 8	syl 17	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
10	9	eleq2d 2875	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝑅 ↔ 𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
11		fveq2 6645	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
12	11	rneqd 5772	. . . . . . . 8 ⊢ (𝑣 = ∅ → ran (𝐹‘𝑣) = ran (𝐹‘∅))
13	12	ineq1d 4138	. . . . . . 7 ⊢ (𝑣 = ∅ → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
14		rneq 5770	. . . . . . . . . . . 12 ⊢ (𝑣 = ∅ → ran 𝑣 = ran ∅)
15		rn0 5760	. . . . . . . . . . . 12 ⊢ ran ∅ = ∅
16	14, 15	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑣 = ∅ → ran 𝑣 = ∅)
17	16	ineq1d 4138	. . . . . . . . . 10 ⊢ (𝑣 = ∅ → (ran 𝑣 ∩ 𝑉) = (∅ ∩ 𝑉))
18		0in 4301	. . . . . . . . . 10 ⊢ (∅ ∩ 𝑉) = ∅
19	17, 18	eqtrdi 2849	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (ran 𝑣 ∩ 𝑉) = ∅)
20	19	iuneq1d 4908	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
21		0iun 4949	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
22	20, 21	eqtrdi 2849	. . . . . . 7 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
23	13, 22	eqeq12d 2814	. . . . . 6 ⊢ (𝑣 = ∅ → ((ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘∅) ∩ 𝑉) = ∅))
24	23	imbi2d 344	. . . . 5 ⊢ (𝑣 = ∅ → ((𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)))
25		fveq2 6645	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
26	25	rneqd 5772	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → ran (𝐹‘𝑣) = ran (𝐹‘𝑦))
27	26	ineq1d 4138	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘𝑦) ∩ 𝑉))
28		rneq 5770	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
29	28	ineq1d 4138	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (ran 𝑣 ∩ 𝑉) = (ran 𝑦 ∩ 𝑉))
30	29	iuneq1d 4908	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
31	27, 30	eqeq12d 2814	. . . . . 6 ⊢ (𝑣 = 𝑦 → ((ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘𝑦) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
32	31	imbi2d 344	. . . . 5 ⊢ (𝑣 = 𝑦 → ((𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑦) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
33		fveq2 6645	. . . . . . . . 9 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹‘𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
34	33	rneqd 5772	. . . . . . . 8 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹‘𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
35	34	ineq1d 4138	. . . . . . 7 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
36		rneq 5770	. . . . . . . . 9 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
37	36	ineq1d 4138	. . . . . . . 8 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣 ∩ 𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
38	37	iuneq1d 4908	. . . . . . 7 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
39	35, 38	eqeq12d 2814	. . . . . 6 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ((ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
40	39	imbi2d 344	. . . . 5 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ((𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
41		fveq2 6645	. . . . . . . . 9 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
42	41	rneqd 5772	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ran (𝐹‘𝑣) = ran (𝐹‘𝑋))
43	42	ineq1d 4138	. . . . . . 7 ⊢ (𝑣 = 𝑋 → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘𝑋) ∩ 𝑉))
44		rneq 5770	. . . . . . . . 9 ⊢ (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
45	44	ineq1d 4138	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (ran 𝑣 ∩ 𝑉) = (ran 𝑋 ∩ 𝑉))
46	45	iuneq1d 4908	. . . . . . 7 ⊢ (𝑣 = 𝑋 → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
47	43, 46	eqeq12d 2814	. . . . . 6 ⊢ (𝑣 = 𝑋 → ((ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
48	47	imbi2d 344	. . . . 5 ⊢ (𝑣 = 𝑋 → ((𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) ↔ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
49	2	mrsub0 32876	. . . . . . . . 9 ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
50	49	rneqd 5772	. . . . . . . 8 ⊢ (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
51	50, 15	eqtrdi 2849	. . . . . . 7 ⊢ (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
52	51	ineq1d 4138	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
53	52, 18	eqtrdi 2849	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
54		uneq1 4083	. . . . . . . 8 ⊢ ((ran (𝐹‘𝑦) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) → ((ran (𝐹‘𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)) = (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
55		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹 ∈ ran 𝑆)
56		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
57	9	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
58	56, 57	eleqtrrd 2893	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ 𝑅)
59		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
60	59	s1cld 13948	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
61	60, 57	eleqtrrd 2893	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ 𝑅)
62	2, 7	mrsubccat 32878	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑦 ∈ 𝑅 ∧ ⟨“𝑧”⟩ ∈ 𝑅) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
63	55, 58, 61, 62	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
64	63	rneqd 5772	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ran ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
65	2, 7	mrsubf 32877	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝑅⟶𝑅)
66	65	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅⟶𝑅)
67	66, 58	ffvelrnd 6829	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘𝑦) ∈ 𝑅)
68	67, 57	eleqtrd 2892	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
69	66, 61	ffvelrnd 6829	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
70	69, 57	eleqtrd 2892	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
71		ccatrn 13934	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹‘𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
72	68, 70, 71	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)) = (ran (𝐹‘𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
73	64, 72	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = (ran (𝐹‘𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)))
74	73	ineq1d 4138	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹‘𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉))
75		indir 4202	. . . . . . . . . 10 ⊢ ((ran (𝐹‘𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹‘𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
76	74, 75	eqtrdi 2849	. . . . . . . . 9 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹‘𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
77		ccatrn 13934	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
78	56, 60, 77	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
79		s1rn 13944	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
80	79	ad2antll 728	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
81	80	uneq2d 4090	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
82	78, 81	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
83	82	ineq1d 4138	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
84		indir 4202	. . . . . . . . . . . . 13 ⊢ ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦 ∩ 𝑉) ∪ ({𝑧} ∩ 𝑉))
85	83, 84	eqtrdi 2849	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∩ 𝑉) ∪ ({𝑧} ∩ 𝑉)))
86	85	iuneq1d 4908	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ ((ran 𝑦 ∩ 𝑉) ∪ ({𝑧} ∩ 𝑉))(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
87		iunxun 4979	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ ((ran 𝑦 ∩ 𝑉) ∪ ({𝑧} ∩ 𝑉))(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
88	86, 87	eqtrdi 2849	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
89		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → 𝑧 ∈ 𝑉)
90	89	snssd 4702	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → {𝑧} ⊆ 𝑉)
91		df-ss 3898	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ⊆ 𝑉 ↔ ({𝑧} ∩ 𝑉) = {𝑧})
92	90, 91	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → ({𝑧} ∩ 𝑉) = {𝑧})
93	92	iuneq1d 4908	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
94		vex 3444	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
95		s1eq 13945	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
96	95	fveq2d 6649	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
97	96	rneqd 5772	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
98	97	ineq1d 4138	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
99	94, 98	iunxsn 4976	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
100	93, 99	eqtrdi 2849	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
101		incom 4128	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ∩ 𝑉) = (𝑉 ∩ {𝑧})
102		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ¬ 𝑧 ∈ 𝑉)
103		disjsn 4607	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑉)
104	102, 103	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (𝑉 ∩ {𝑧}) = ∅)
105	101, 104	syl5eq 2845	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ({𝑧} ∩ 𝑉) = ∅)
106	105	iuneq1d 4908	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
107	55	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → 𝐹 ∈ ran 𝑆)
108		eldif 3891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) ↔ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧 ∈ 𝑉))
109	108	biimpri 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧 ∈ 𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
110	59, 109	sylan 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
111		difun2 4387	. . . . . . . . . . . . . . . . . . 19 ⊢ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
112	110, 111	eleqtrdi 2900	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉))
113	2, 7, 6, 5	mrsubcn 32879	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉)) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
114	107, 112, 113	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
115	114	rneqd 5772	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ran (𝐹‘⟨“𝑧”⟩) = ran ⟨“𝑧”⟩)
116	80	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
117	115, 116	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
118	117	ineq1d 4138	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
119	118, 105	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
120	21, 106, 119	3eqtr4a 2859	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
121	100, 120	pm2.61dan 812	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
122	121	uneq2d 4090	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) = (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
123	88, 122	eqtrd 2833	. . . . . . . . 9 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)))
124	76, 123	eqeq12d 2814	. . . . . . . 8 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ((ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ ((ran (𝐹‘𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)) = (∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))))
125	54, 124	syl5ibr 249	. . . . . . 7 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ((ran (𝐹‘𝑦) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
126	125	expcom 417	. . . . . 6 ⊢ ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉)) → (𝐹 ∈ ran 𝑆 → ((ran (𝐹‘𝑦) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
127	126	a2d 29	. . . . 5 ⊢ ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉)) → ((𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑦) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)) → (𝐹 ∈ ran 𝑆 → (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉) = ∪ 𝑥 ∈ (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))))
128	24, 32, 40, 48, 53, 127	wrdind 14075	. . . 4 ⊢ (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
129	128	com12 32	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
130	10, 129	sylbid 243	. 2 ⊢ (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝑅 → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
131	130	imp 410	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅) → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 ∪ ciun 4881 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Word cword 13857 ++ cconcat 13913 ⟨“cs1 13940 mCNcmcn 32820 mVRcmvar 32821 mRExcmrex 32826 mRSubstcmrsub 32830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-frmd 18006 df-mrex 32846 df-mrsub 32850

This theorem is referenced by: msubvrs 32920

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