Theorem msubff1 36143

Description: When restricted to complete mappings, the substitution-producing function is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
msubff1.v	⊢ 𝑉 = (mVR‘𝑇)
msubff1.r	⊢ 𝑅 = (mREx‘𝑇)
msubff1.s	⊢ 𝑆 = (mSubst‘𝑇)
msubff1.e	⊢ 𝐸 = (mEx‘𝑇)

Assertion

Ref	Expression
msubff1	⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸))

Proof of Theorem msubff1

Dummy variables 𝑓 𝑔 𝑟 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubff1.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
2		msubff1.r	. . . 4 ⊢ 𝑅 = (mREx‘𝑇)
3		msubff1.s	. . . 4 ⊢ 𝑆 = (mSubst‘𝑇)
4		msubff1.e	. . . 4 ⊢ 𝐸 = (mEx‘𝑇)
5	1, 2, 3, 4	msubff 36117	. . 3 ⊢ (𝑇 ∈ mFS → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
6		mapsspm 8887	. . . 4 ⊢ (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉)
7	6	a1i 11	. . 3 ⊢ (𝑇 ∈ mFS → (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉))
8	5, 7	fssresd 6746	. 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)⟶(𝐸 ↑m 𝐸))
9		eqid 2762	. . . . . . . . . . . . 13 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	1, 2, 9	mrsubff 36099	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
11	10	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
12		simplrl 789	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑓 ∈ (𝑅 ↑m 𝑉))
13	6, 12	sselid 3932	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑓 ∈ (𝑅 ↑pm 𝑉))
14	11, 13	ffvelcdmd 7082	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅))
15		elmapi 8852	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
16		ffn 6706	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 → ((mRSubst‘𝑇)‘𝑓) Fn 𝑅)
17	14, 15, 16	3syl 19	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑓) Fn 𝑅)
18		simplrr 790	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑔 ∈ (𝑅 ↑m 𝑉))
19	6, 18	sselid 3932	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑔 ∈ (𝑅 ↑pm 𝑉))
20	11, 19	ffvelcdmd 7082	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅 ↑m 𝑅))
21		elmapi 8852	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅)
22		ffn 6706	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅 → ((mRSubst‘𝑇)‘𝑔) Fn 𝑅)
23	20, 21, 22	3syl 19	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑔) Fn 𝑅)
24		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → (𝑆‘𝑓) = (𝑆‘𝑔))
25	24	fveq1d 6884	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ((𝑆‘𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ((𝑆‘𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))
26	12	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑓 ∈ (𝑅 ↑m 𝑉))
27		elmapi 8852	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → 𝑓:𝑉⟶𝑅)
28	26, 27	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑓:𝑉⟶𝑅)
29		ssidd 3957	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑉 ⊆ 𝑉)
30		eqid 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
31		eqid 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (mType‘𝑇) = (mType‘𝑇)
32	1, 30, 31	mtyf2 36138	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
33	32	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
34		simplrl 789	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑣 ∈ 𝑉)
35	33, 34	ffvelcdmd 7082	. . . . . . . . . . . . . . 15 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
36		opelxpi 5696	. . . . . . . . . . . . . . 15 ⊢ ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟 ∈ 𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
37	35, 36	sylancom 600	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
38	30, 4, 2	mexval 36089	. . . . . . . . . . . . . 14 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
39	37, 38	eleqtrrdi 2873	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸)
40	1, 2, 3, 4, 9	msubval 36112	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸) → ((𝑆‘𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
41	28, 29, 39, 40	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ((𝑆‘𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
42	18	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑔 ∈ (𝑅 ↑m 𝑉))
43		elmapi 8852	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → 𝑔:𝑉⟶𝑅)
44	42, 43	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑔:𝑉⟶𝑅)
45	1, 2, 3, 4, 9	msubval 36112	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸) → ((𝑆‘𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
46	44, 29, 39, 45	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ((𝑆‘𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
47	25, 41, 46	3eqtr3d 2805	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
48		fvex 6895	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49		fvex 6895	. . . . . . . . . . . . 13 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
50	48, 49	opth 5456	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ ↔ ((1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))))
51	50	simprbi 503	. . . . . . . . . . 11 ⊢ (⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)))
52	47, 51	syl 18	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)))
53		fvex 6895	. . . . . . . . . . . 12 ⊢ ((mType‘𝑇)‘𝑣) ∈ V
54		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
55	53, 54	op2nd 7999	. . . . . . . . . . 11 ⊢ (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
56	55	fveq2i 6885	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
57	55	fveq2i 6885	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
58	52, 56, 57	3eqtr3g 2820	. . . . . . . . 9 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘𝑟) = (((mRSubst‘𝑇)‘𝑔)‘𝑟))
59	17, 23, 58	eqfnfvd 7029	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
60	1, 2, 9	mrsubff1 36101	. . . . . . . . . . 11 ⊢ (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅))
61		f1fveq 7263	. . . . . . . . . . 11 ⊢ ((((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅) ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
62	60, 61	sylan 592	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
63		fvres 6901	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64		fvres 6901	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
65	63, 64	eqeqan12d 2776	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) → ((((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
66	65	adantl 487	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
67	62, 66	bitr3d 284	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
68	67	adantr 486	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
69	59, 68	mpbird 260	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑓 = 𝑔)
70	69	fveq1d 6884	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → (𝑓‘𝑣) = (𝑔‘𝑣))
71	70	expr 462	. . . . 5 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑣 ∈ 𝑉) → ((𝑆‘𝑓) = (𝑆‘𝑔) → (𝑓‘𝑣) = (𝑔‘𝑣)))
72	71	ralrimdva 3164	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → ((𝑆‘𝑓) = (𝑆‘𝑔) → ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
73		fvres 6901	. . . . . 6 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = (𝑆‘𝑓))
74		fvres 6901	. . . . . 6 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) = (𝑆‘𝑔))
75	73, 74	eqeqan12d 2776	. . . . 5 ⊢ ((𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) → (((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) ↔ (𝑆‘𝑓) = (𝑆‘𝑔)))
76	75	adantl 487	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → (((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) ↔ (𝑆‘𝑓) = (𝑆‘𝑔)))
77		ffn 6706	. . . . . . 7 ⊢ (𝑓:𝑉⟶𝑅 → 𝑓 Fn 𝑉)
78		ffn 6706	. . . . . . 7 ⊢ (𝑔:𝑉⟶𝑅 → 𝑔 Fn 𝑉)
79		eqfnfv 7026	. . . . . . 7 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
80	77, 78, 79	syl2an 608	. . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑔:𝑉⟶𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
81	27, 43, 80	syl2an 608	. . . . 5 ⊢ ((𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
82	81	adantl 487	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
83	72, 76, 82	3imtr4d 297	. . 3 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → (((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) → 𝑓 = 𝑔))
84	83	ralrimivva 3207	. 2 ⊢ (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅 ↑m 𝑉)∀𝑔 ∈ (𝑅 ↑m 𝑉)(((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) → 𝑓 = 𝑔))
85		dff13 7255	. 2 ⊢ ((𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸) ↔ ((𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)⟶(𝐸 ↑m 𝐸) ∧ ∀𝑓 ∈ (𝑅 ↑m 𝑉)∀𝑔 ∈ (𝑅 ↑m 𝑉)(((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) → 𝑓 = 𝑔)))
86	8, 84, 85	sylanbrc 595	1 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  ⟨cop 4593   × cxp 5657   ↾ cres 5661   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989   ↑_m cmap 8830   ↑_pm cpm 8831  mVRcmvar 36048  mTypecmty 36049  mTCcmtc 36051  mRExcmrex 36053  mExcmex 36054  mRSubstcmrsub 36057  mSubstcmsub 36058  mFScmfs 36063

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-seq 14070  df-hash 14399  df-word 14583  df-concat 14640  df-s1 14667  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-0g 17532  df-gsum 17533  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-frmd 18964  df-mrex 36073  df-mex 36074  df-mrsub 36077  df-msub 36078  df-mfs 36083

This theorem is used by:  msubff1o  36144