mrsubvrs - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubvrs		Structured version Visualization version GIF version

Theorem mrsubvrs 35750

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 4268	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2		mrsubco.s	. . . . . . 7 ⊢ 𝑆 = (mRSubst‘𝑇)
3	2	rnfvprc 6821	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
4	1, 3	nsyl2 141	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
5		eqid 2739	. . . . . 6 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
6		mrsubvrs.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
7		mrsubvrs.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
8	5, 6, 7	mrexval 35729	. . . . 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 2825	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝑅 ↔ 𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉)))
11		fveq2 6827	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
12	11	rneqd 5880	. . . . . . . 8 ⊢ (𝑣 = ∅ → ran (𝐹‘𝑣) = ran (𝐹‘∅))
13	12	ineq1d 4148	. . . . . . 7 ⊢ (𝑣 = ∅ → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘∅) ∩ 𝑉))
14		rneq 5878	. . . . . . . . . . . 12 ⊢ (𝑣 = ∅ → ran 𝑣 = ran ∅)
15		rn0 5868	. . . . . . . . . . . 12 ⊢ ran ∅ = ∅
16	14, 15	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (𝑣 = ∅ → ran 𝑣 = ∅)
17	16	ineq1d 4148	. . . . . . . . . 10 ⊢ (𝑣 = ∅ → (ran 𝑣 ∩ 𝑉) = (∅ ∩ 𝑉))
18		0in 4325	. . . . . . . . . 10 ⊢ (∅ ∩ 𝑉) = ∅
19	17, 18	eqtrdi 2790	. . . . . . . . 9 ⊢ (𝑣 = ∅ → (ran 𝑣 ∩ 𝑉) = ∅)
20	19	iuneq1d 4949	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
21		0iun 4992	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ ∅ (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅
22	20, 21	eqtrdi 2790	. . . . . . 7 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∅)
23	13, 22	eqeq12d 2755	. . . . . 6 ⊢ (𝑣 = ∅ → ((ran (𝐹‘𝑣) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) ↔ (ran (𝐹‘∅) ∩ 𝑉) = ∅))
24	23	imbi2d 341	. . . . 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 6827	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
26	25	rneqd 5880	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → ran (𝐹‘𝑣) = ran (𝐹‘𝑦))
27	26	ineq1d 4148	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘𝑦) ∩ 𝑉))
28		rneq 5878	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ran 𝑣 = ran 𝑦)
29	28	ineq1d 4148	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (ran 𝑣 ∩ 𝑉) = (ran 𝑦 ∩ 𝑉))
30	29	iuneq1d 4949	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑦 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
31	27, 30	eqeq12d 2755	. . . . . 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 341	. . . . 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 6827	. . . . . . . . 9 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (𝐹‘𝑣) = (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
34	33	rneqd 5880	. . . . . . . 8 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran (𝐹‘𝑣) = ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)))
35	34	ineq1d 4148	. . . . . . 7 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) ∩ 𝑉))
36		rneq 5878	. . . . . . . . 9 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → ran 𝑣 = ran (𝑦 ++ ⟨“𝑧”⟩))
37	36	ineq1d 4148	. . . . . . . 8 ⊢ (𝑣 = (𝑦 ++ ⟨“𝑧”⟩) → (ran 𝑣 ∩ 𝑉) = (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉))
38	37	iuneq1d 4949	. . . . . . 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 2755	. . . . . 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 341	. . . . 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 6827	. . . . . . . . 9 ⊢ (𝑣 = 𝑋 → (𝐹‘𝑣) = (𝐹‘𝑋))
42	41	rneqd 5880	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → ran (𝐹‘𝑣) = ran (𝐹‘𝑋))
43	42	ineq1d 4148	. . . . . . 7 ⊢ (𝑣 = 𝑋 → (ran (𝐹‘𝑣) ∩ 𝑉) = (ran (𝐹‘𝑋) ∩ 𝑉))
44		rneq 5878	. . . . . . . . 9 ⊢ (𝑣 = 𝑋 → ran 𝑣 = ran 𝑋)
45	44	ineq1d 4148	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (ran 𝑣 ∩ 𝑉) = (ran 𝑋 ∩ 𝑉))
46	45	iuneq1d 4949	. . . . . . 7 ⊢ (𝑣 = 𝑋 → ∪ 𝑥 ∈ (ran 𝑣 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉))
47	43, 46	eqeq12d 2755	. . . . . 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 341	. . . . 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 35744	. . . . . . . . 9 ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅)
50	49	rneqd 5880	. . . . . . . 8 ⊢ (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ran ∅)
51	50, 15	eqtrdi 2790	. . . . . . 7 ⊢ (𝐹 ∈ ran 𝑆 → ran (𝐹‘∅) = ∅)
52	51	ineq1d 4148	. . . . . 6 ⊢ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = (∅ ∩ 𝑉))
53	52, 18	eqtrdi 2790	. . . . 5 ⊢ (𝐹 ∈ ran 𝑆 → (ran (𝐹‘∅) ∩ 𝑉) = ∅)
54		uneq1 4091	. . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹 ∈ ran 𝑆)
56		simprl 776	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
57	9	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
58	56, 57	eleqtrrd 2842	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑦 ∈ 𝑅)
59		simprr 778	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))
60	59	s1cld 14557	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
61	60, 57	eleqtrrd 2842	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ⟨“𝑧”⟩ ∈ 𝑅)
62	2, 7	mrsubccat 35746	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑦 ∈ 𝑅 ∧ ⟨“𝑧”⟩ ∈ 𝑅) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
63	55, 58, 61, 62	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘(𝑦 ++ ⟨“𝑧”⟩)) = ((𝐹‘𝑦) ++ (𝐹‘⟨“𝑧”⟩)))
64	63	rneqd 5880	. . . . . . . . . . . 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 35745	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝑅⟶𝑅)
66	65	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → 𝐹:𝑅⟶𝑅)
67	66, 58	ffvelcdmd 7026	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘𝑦) ∈ 𝑅)
68	67, 57	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘𝑦) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
69	66, 61	ffvelcdmd 7026	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ 𝑅)
70	69, 57	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (𝐹‘⟨“𝑧”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
71		ccatrn 14543	. . . . . . . . . . . . 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 590	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 4148	. . . . . . . . . 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 4214	. . . . . . . . . 10 ⊢ ((ran (𝐹‘𝑦) ∪ ran (𝐹‘⟨“𝑧”⟩)) ∩ 𝑉) = ((ran (𝐹‘𝑦) ∩ 𝑉) ∪ (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
76	74, 75	eqtrdi 2790	. . . . . . . . 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 14543	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ ⟨“𝑧”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉)) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
78	56, 60, 77	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ ran ⟨“𝑧”⟩))
79		s1rn 14553	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
80	79	ad2antll 735	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran ⟨“𝑧”⟩ = {𝑧})
81	80	uneq2d 4098	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran 𝑦 ∪ ran ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
82	78, 81	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ran (𝑦 ++ ⟨“𝑧”⟩) = (ran 𝑦 ∪ {𝑧}))
83	82	ineq1d 4148	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → (ran (𝑦 ++ ⟨“𝑧”⟩) ∩ 𝑉) = ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉))
84		indir 4214	. . . . . . . . . . . . 13 ⊢ ((ran 𝑦 ∪ {𝑧}) ∩ 𝑉) = ((ran 𝑦 ∩ 𝑉) ∪ ({𝑧} ∩ 𝑉))
85	83, 84	eqtrdi 2790	. . . . . . . . . . . 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 4949	. . . . . . . . . . 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 5023	. . . . . . . . . . 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 2790	. . . . . . . . . 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 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → 𝑧 ∈ 𝑉)
90	89	snssd 4718	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → {𝑧} ⊆ 𝑉)
91		dfss2 3901	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ⊆ 𝑉 ↔ ({𝑧} ∩ 𝑉) = {𝑧})
92	90, 91	sylib 219	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ 𝑧 ∈ 𝑉) → ({𝑧} ∩ 𝑉) = {𝑧})
93	92	iuneq1d 4949	. . . . . . . . . . . . 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 3435	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
95		s1eq 14554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ⟨“𝑥”⟩ = ⟨“𝑧”⟩)
96	95	fveq2d 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹‘⟨“𝑥”⟩) = (𝐹‘⟨“𝑧”⟩))
97	96	rneqd 5880	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ran (𝐹‘⟨“𝑥”⟩) = ran (𝐹‘⟨“𝑧”⟩))
98	97	ineq1d 4148	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
99	94, 98	iunxsn 5020	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ {𝑧} (ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉)
100	93, 99	eqtrdi 2790	. . . . . . . . . . . 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 4138	. . . . . . . . . . . . . . 15 ⊢ ({𝑧} ∩ 𝑉) = (𝑉 ∩ {𝑧})
102		disjsn 4643	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑉)
103	102	bilanri 507	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (𝑉 ∩ {𝑧}) = ∅)
104	101, 103	eqtrid 2786	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ({𝑧} ∩ 𝑉) = ∅)
105	104	iuneq1d 4949	. . . . . . . . . . . . 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))
106	55	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → 𝐹 ∈ ran 𝑆)
107		eldif 3893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) ↔ (𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧 ∈ 𝑉))
108	107	biimpri 229	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉) ∧ ¬ 𝑧 ∈ 𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
109	59, 108	sylan 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → 𝑧 ∈ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉))
110		difun2 4409	. . . . . . . . . . . . . . . . . . 19 ⊢ (((mCN‘𝑇) ∪ 𝑉) ∖ 𝑉) = ((mCN‘𝑇) ∖ 𝑉)
111	109, 110	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉))
112	2, 7, 6, 5	mrsubcn 35747	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑧 ∈ ((mCN‘𝑇) ∖ 𝑉)) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
113	106, 111, 112	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (𝐹‘⟨“𝑧”⟩) = ⟨“𝑧”⟩)
114	113	rneqd 5880	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ran (𝐹‘⟨“𝑧”⟩) = ran ⟨“𝑧”⟩)
115	80	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ran ⟨“𝑧”⟩ = {𝑧})
116	114, 115	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → ran (𝐹‘⟨“𝑧”⟩) = {𝑧})
117	116	ineq1d 4148	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ({𝑧} ∩ 𝑉))
118	117, 104	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) ∧ ¬ 𝑧 ∈ 𝑉) → (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉) = ∅)
119	21, 105, 118	3eqtr4a 2800	. . . . . . . . . . . 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))
120	100, 119	pm2.61dan 818	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝑦 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ 𝑧 ∈ ((mCN‘𝑇) ∪ 𝑉))) → ∪ 𝑥 ∈ ({𝑧} ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉) = (ran (𝐹‘⟨“𝑧”⟩) ∩ 𝑉))
121	120	uneq2d 4098	. . . . . . . . . 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)))
122	88, 121	eqtrd 2774	. . . . . . . . 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)))
123	76, 122	eqeq12d 2755	. . . . . . . 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))))
124	54, 123	imbitrrid 247	. . . . . . 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)))
125	124	expcom 414	. . . . . 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))))
126	125	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))))
127	24, 32, 40, 48, 53, 126	wrdind 14675	. . . 4 ⊢ (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (𝐹 ∈ ran 𝑆 → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
128	127	com12 32	. . 3 ⊢ (𝐹 ∈ ran 𝑆 → (𝑋 ∈ Word ((mCN‘𝑇) ∪ 𝑉) → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
129	10, 128	sylbid 241	. 2 ⊢ (𝐹 ∈ ran 𝑆 → (𝑋 ∈ 𝑅 → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘⟨“𝑥”⟩) ∩ 𝑉)))
130	129	imp 407	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 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 {csn 4555 ∪ ciun 4921 ran crn 5619 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 Word cword 14466 ++ cconcat 14523 ⟨“cs1 14549 mCNcmcn 35688 mVRcmvar 35689 mRExcmrex 35694 mRSubstcmrsub 35698

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-word 14467 df-lsw 14516 df-concat 14524 df-s1 14550 df-substr 14595 df-pfx 14625 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-0g 17395 df-gsum 17396 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-frmd 18808 df-mrex 35714 df-mrsub 35718

This theorem is referenced by: msubvrs 35788

W3C validator