Theorem msubff1 35750

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 35724	. . 3 ⊢ (𝑇 ∈ mFS → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))
6		mapsspm 8814	. . . 4 ⊢ (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉)
7	6	a1i 11	. . 3 ⊢ (𝑇 ∈ mFS → (𝑅 ↑m 𝑉) ⊆ (𝑅 ↑pm 𝑉))
8	5, 7	fssresd 6701	. 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)⟶(𝐸 ↑m 𝐸))
9		eqid 2736	. . . . . . . . . . . . 13 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	1, 2, 9	mrsubff 35706	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅))
11	10	ad2antrr 726	. . . . . . . . . . 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 776	. . . . . . . . . . . 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 3931	. . . . . . . . . . 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 7030	. . . . . . . . . 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 8786	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
16		ffn 6662	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 → ((mRSubst‘𝑇)‘𝑓) Fn 𝑅)
17	14, 15, 16	3syl 18	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑓) Fn 𝑅)
18		simplrr 777	. . . . . . . . . . . 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 3931	. . . . . . . . . . 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 7030	. . . . . . . . . 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 8786	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅)
22		ffn 6662	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅 → ((mRSubst‘𝑇)‘𝑔) Fn 𝑅)
23	20, 21, 22	3syl 18	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → ((mRSubst‘𝑇)‘𝑔) Fn 𝑅)
24		simplrr 777	. . . . . . . . . . . . 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 6836	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 8786	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → 𝑓:𝑉⟶𝑅)
28	26, 27	syl 17	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
31		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (mType‘𝑇) = (mType‘𝑇)
32	1, 30, 31	mtyf2 35745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
33	32	ad3antrrr 730	. . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 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 7030	. . . . . . . . . . . . . . 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 5661	. . . . . . . . . . . . . . 15 ⊢ ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟 ∈ 𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
37	35, 36	sylancom 588	. . . . . . . . . . . . . 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 35696	. . . . . . . . . . . . . 14 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
39	37, 38	eleqtrrdi 2847	. . . . . . . . . . . . 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 35719	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 8786	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → 𝑔:𝑉⟶𝑅)
44	42, 43	syl 17	. . . . . . . . . . . . 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 35719	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 6847	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49		fvex 6847	. . . . . . . . . . . . 13 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
50	48, 49	opth 5424	. . . . . . . . . . . 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 496	. . . . . . . . . . 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 17	. . . . . . . . . 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 6847	. . . . . . . . . . . 12 ⊢ ((mType‘𝑇)‘𝑣) ∈ V
54		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
55	53, 54	op2nd 7942	. . . . . . . . . . 11 ⊢ (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
56	55	fveq2i 6837	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
57	55	fveq2i 6837	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
58	52, 56, 57	3eqtr3g 2794	. . . . . . . . 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 6979	. . . . . . . 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 35708	. . . . . . . . . . 11 ⊢ (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅))
61		f1fveq 7208	. . . . . . . . . . 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 580	. . . . . . . . . 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 6853	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64		fvres 6853	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
65	63, 64	eqeqan12d 2750	. . . . . . . . . . 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 481	. . . . . . . . . 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 281	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
68	67	adantr 480	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑓 = 𝑔)
70	69	fveq1d 6836	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → (𝑓‘𝑣) = (𝑔‘𝑣))
71	70	expr 456	. . . . 5 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑣 ∈ 𝑉) → ((𝑆‘𝑓) = (𝑆‘𝑔) → (𝑓‘𝑣) = (𝑔‘𝑣)))
72	71	ralrimdva 3136	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → ((𝑆‘𝑓) = (𝑆‘𝑔) → ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
73		fvres 6853	. . . . . 6 ⊢ (𝑓 ∈ (𝑅 ↑m 𝑉) → ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑓) = (𝑆‘𝑓))
74		fvres 6853	. . . . . 6 ⊢ (𝑔 ∈ (𝑅 ↑m 𝑉) → ((𝑆 ↾ (𝑅 ↑m 𝑉))‘𝑔) = (𝑆‘𝑔))
75	73, 74	eqeqan12d 2750	. . . . 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 481	. . . 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 6662	. . . . . . 7 ⊢ (𝑓:𝑉⟶𝑅 → 𝑓 Fn 𝑉)
78		ffn 6662	. . . . . . 7 ⊢ (𝑔:𝑉⟶𝑅 → 𝑔 Fn 𝑉)
79		eqfnfv 6976	. . . . . . 7 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
80	77, 78, 79	syl2an 596	. . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑔:𝑉⟶𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
81	27, 43, 80	syl2an 596	. . . . 5 ⊢ ((𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
82	81	adantl 481	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑m 𝑉))) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
83	72, 76, 82	3imtr4d 294	. . 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 3179	. 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 7200	. 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 583	1 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901  ⟨cop 4586   × cxp 5622   ↾ cres 5626   Fn wfn 6487  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8763   ↑_pm cpm 8764  mVRcmvar 35655  mTypecmty 35656  mTCcmtc 35658  mRExcmrex 35660  mExcmex 35661  mRSubstcmrsub 35664  mSubstcmsub 35665  mFScmfs 35670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-word 14437  df-concat 14494  df-s1 14520  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-gsum 17362  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-frmd 18774  df-mrex 35680  df-mex 35681  df-mrsub 35684  df-msub 35685  df-mfs 35690

This theorem is referenced by:  msubff1o  35751