Theorem msubff1 36029

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 36003	. . 3 ⊢ (𝑇 ∈ mFS → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
6		mapsspm 8875	. . . 4 ⊢ (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉)
7	6	a1i 11	. . 3 ⊢ (𝑇 ∈ mFS → (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉))
8	5, 7	fssresd 6747	. 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)⟶(𝐸 ↑m 𝐸))
9		eqid 2763	. . . . . . . . . . . . 13 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	1, 2, 9	mrsubff 35985	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
11	10	ad2antrr 738	. . . . . . . . . . 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 788	. . . . . . . . . . . 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 3936	. . . . . . . . . . 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 8847	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
16		ffn 6707	. . . . . . . . . 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 789	. . . . . . . . . . . 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 3936	. . . . . . . . . . 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 8847	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅)
22		ffn 6707	. . . . . . . . . 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 789	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 8847	. . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑉 ⊆ 𝑉)
30		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
31		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (mType‘𝑇) = (mType‘𝑇)
32	1, 30, 31	mtyf2 36024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
33	32	ad3antrrr 742	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . 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 5700	. . . . . . . . . . . . . . 15 ⊢ ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟 ∈ 𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
37	35, 36	sylancom 599	. . . . . . . . . . . . . 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 35975	. . . . . . . . . . . . . 14 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
39	37, 38	eleqtrrdi 2874	. . . . . . . . . . . . 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 35998	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 8847	. . . . . . . . . . . . . 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 35998	. . . . . . . . . . . . 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 2806	. . . . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49		fvex 6896	. . . . . . . . . . . . 13 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
50	48, 49	opth 5460	. . . . . . . . . . . 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 502	. . . . . . . . . . 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 6896	. . . . . . . . . . . 12 ⊢ ((mType‘𝑇)‘𝑣) ∈ V
54		vex 3459	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
55	53, 54	op2nd 7996	. . . . . . . . . . 11 ⊢ (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
56	55	fveq2i 6886	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
57	55	fveq2i 6886	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
58	52, 56, 57	3eqtr3g 2821	. . . . . . . . 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 7030	. . . . . . . 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 35987	. . . . . . . . . . 11 ⊢ (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅))
61		f1fveq 7262	. . . . . . . . . . 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 591	. . . . . . . . . 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 6902	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64		fvres 6902	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
65	63, 64	eqeqan12d 2777	. . . . . . . . . . 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 486	. . . . . . . . . 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 485	. . . . . . . 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 6885	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → (𝑓‘𝑣) = (𝑔‘𝑣))
71	70	expr 461	. . . . 5 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑣 ∈ 𝑉) → ((𝑆‘𝑓) = (𝑆‘𝑔) → (𝑓‘𝑣) = (𝑔‘𝑣)))
72	71	ralrimdva 3165	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → ((𝑆‘𝑓) = (𝑆‘𝑔) → ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
73		fvres 6902	. . . . . 6 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = (𝑆‘𝑓))
74		fvres 6902	. . . . . 6 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) = (𝑆‘𝑔))
75	73, 74	eqeqan12d 2777	. . . . 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 486	. . . 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 6707	. . . . . . 7 ⊢ (𝑓:𝑉⟶𝑅 → 𝑓 Fn 𝑉)
78		ffn 6707	. . . . . . 7 ⊢ (𝑔:𝑉⟶𝑅 → 𝑔 Fn 𝑉)
79		eqfnfv 7027	. . . . . . 7 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
80	77, 78, 79	syl2an 607	. . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑔:𝑉⟶𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
81	27, 43, 80	syl2an 607	. . . . 5 ⊢ ((𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
82	81	adantl 486	. . . 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 3208	. 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 7254	. 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 594	1 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3906  ⟨cop 4596   × cxp 5661   ↾ cres 5665   Fn wfn 6533  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986   ↑_m cmap 8825   ↑_pm cpm 8826  mVRcmvar 35934  mTypecmty 35935  mTCcmtc 35937  mRExcmrex 35939  mExcmex 35940  mRSubstcmrsub 35943  mSubstcmsub 35944  mFScmfs 35949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-struct 17208  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-submnd 18843  df-frmd 18909  df-mrex 35959  df-mex 35960  df-mrsub 35963  df-msub 35964  df-mfs 35969

This theorem is referenced by:  msubff1o  36030