msubff1 - Mathbox for Mario Carneiro

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Theorem msubff1 34535

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 34509	. . 3 ⊢ (𝑇 ∈ mFS → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))
6		mapsspm 8866	. . . 4 ⊢ (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉)
7	6	a1i 11	. . 3 ⊢ (𝑇 ∈ mFS → (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉))
8	5, 7	fssresd 6755	. 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)⟶(𝐸 ↑_m 𝐸))
9		eqid 2732	. . . . . . . . . . . . 13 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	1, 2, 9	mrsubff 34491	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
11	10	ad2antrr 724	. . . . . . . . . . 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 775	. . . . . . . . . . . 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 3979	. . . . . . . . . . 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 7084	. . . . . . . . . 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 8839	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
16		ffn 6714	. . . . . . . . . 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 776	. . . . . . . . . . . 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 3979	. . . . . . . . . . 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 7084	. . . . . . . . . 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 8839	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅 ↑_m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅)
22		ffn 6714	. . . . . . . . . 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 776	. . . . . . . . . . . . 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 6890	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 8839	. . . . . . . . . . . . . 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 4004	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑉 ⊆ 𝑉)
30		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
31		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (mType‘𝑇) = (mType‘𝑇)
32	1, 30, 31	mtyf2 34530	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
33	32	ad3antrrr 728	. . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . 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 5712	. . . . . . . . . . . . . . 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 34481	. . . . . . . . . . . . . 14 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
39	37, 38	eleqtrrdi 2844	. . . . . . . . . . . . 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 34504	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸) → ((𝑆‘𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
41	28, 29, 39, 40	syl3anc 1371	. . . . . . . . . . . 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⟩) = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
42	18	adantr 481	. . . . . . . . . . . . . 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 8839	. . . . . . . . . . . . . 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 34504	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝑉⟶𝑅 ∧ 𝑉 ⊆ 𝑉 ∧ ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸) → ((𝑆‘𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
46	44, 29, 39, 45	syl3anc 1371	. . . . . . . . . . . 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⟩) = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
47	25, 41, 46	3eqtr3d 2780	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
48		fvex 6901	. . . . . . . . . . . . 13 ⊢ (1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49		fvex 6901	. . . . . . . . . . . . 13 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
50	48, 49	opth 5475	. . . . . . . . . . . 12 ⊢ (⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ ↔ ((1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = (1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))))
51	50	simprbi 497	. . . . . . . . . . 11 ⊢ (⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ = ⟨(1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩ → (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((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)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)))
53		fvex 6901	. . . . . . . . . . . 12 ⊢ ((mType‘𝑇)‘𝑣) ∈ V
54		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
55	53, 54	op2nd 7980	. . . . . . . . . . 11 ⊢ (2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
56	55	fveq2i 6891	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
57	55	fveq2i 6891	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
58	52, 56, 57	3eqtr3g 2795	. . . . . . . . 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 7032	. . . . . . . 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 34493	. . . . . . . . . . 11 ⊢ (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1→(𝑅 ↑_m 𝑅))
61		f1fveq 7257	. . . . . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑_m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64		fvres 6907	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑_m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
65	63, 64	eqeqan12d 2746	. . . . . . . . . . 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 482	. . . . . . . . . 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 280	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
68	67	adantr 481	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → 𝑓 = 𝑔)
70	69	fveq1d 6890	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) → (𝑓‘𝑣) = (𝑔‘𝑣))
71	70	expr 457	. . . . 5 ⊢ (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) ∧ 𝑣 ∈ 𝑉) → ((𝑆‘𝑓) = (𝑆‘𝑔) → (𝑓‘𝑣) = (𝑔‘𝑣)))
72	71	ralrimdva 3154	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) → ((𝑆‘𝑓) = (𝑆‘𝑔) → ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
73		fvres 6907	. . . . . 6 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝑉) → ((𝑆 ↾ (𝑅 ↑_m 𝑉))‘𝑓) = (𝑆‘𝑓))
74		fvres 6907	. . . . . 6 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → ((𝑆 ↾ (𝑅 ↑_m 𝑉))‘𝑔) = (𝑆‘𝑔))
75	73, 74	eqeqan12d 2746	. . . . 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 482	. . . 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 6714	. . . . . . 7 ⊢ (𝑓:𝑉⟶𝑅 → 𝑓 Fn 𝑉)
78		ffn 6714	. . . . . . 7 ⊢ (𝑔:𝑉⟶𝑅 → 𝑔 Fn 𝑉)
79		eqfnfv 7029	. . . . . . 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 482	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
83	72, 76, 82	3imtr4d 293	. . 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 3200	. 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 7250	. 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3947 ⟨cop 4633 × cxp 5673 ↾ cres 5677 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8816 ↑_pm cpm 8817 mVRcmvar 34440 mTypecmty 34441 mTCcmtc 34443 mRExcmrex 34445 mExcmex 34446 mRSubstcmrsub 34449 mSubstcmsub 34450 mFScmfs 34455

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-frmd 18726 df-mrex 34465 df-mex 34466 df-mrsub 34469 df-msub 34470 df-mfs 34475

This theorem is referenced by: msubff1o 34536

W3C validator