msubff1 - Mathbox for Mario Carneiro

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Theorem msubff1 35012

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 34986	. . 3 ⊢ (𝑇 ∈ mFS → 𝑆:(𝑅 ↑_pm 𝑉)⟶(𝐸 ↑_m 𝐸))
6		mapsspm 8876	. . . 4 ⊢ (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉)
7	6	a1i 11	. . 3 ⊢ (𝑇 ∈ mFS → (𝑅 ↑_m 𝑉) ⊆ (𝑅 ↑_pm 𝑉))
8	5, 7	fssresd 6758	. 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)⟶(𝐸 ↑_m 𝐸))
9		eqid 2731	. . . . . . . . . . . . 13 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
10	1, 2, 9	mrsubff 34968	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅 ↑_pm 𝑉)⟶(𝑅 ↑_m 𝑅))
11	10	ad2antrr 723	. . . . . . . . . . 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 774	. . . . . . . . . . . 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 3980	. . . . . . . . . . 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 7087	. . . . . . . . . 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 8849	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑_m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅)
16		ffn 6717	. . . . . . . . . 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 775	. . . . . . . . . . . 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 3980	. . . . . . . . . . 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 7087	. . . . . . . . . 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 8849	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅 ↑_m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅⟶𝑅)
22		ffn 6717	. . . . . . . . . 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 775	. . . . . . . . . . . . 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 6893	. . . . . . . . . . . 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 8849	. . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) ∧ (𝑣 ∈ 𝑉 ∧ (𝑆‘𝑓) = (𝑆‘𝑔))) ∧ 𝑟 ∈ 𝑅) → 𝑉 ⊆ 𝑉)
30		eqid 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
31		eqid 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (mType‘𝑇) = (mType‘𝑇)
32	1, 30, 31	mtyf2 35007	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
33	32	ad3antrrr 727	. . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 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 7087	. . . . . . . . . . . . . . 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 5713	. . . . . . . . . . . . . . 15 ⊢ ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟 ∈ 𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
37	35, 36	sylancom 587	. . . . . . . . . . . . . 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 34958	. . . . . . . . . . . . . 14 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅)
39	37, 38	eleqtrrdi 2843	. . . . . . . . . . . . 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 34981	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 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 8849	. . . . . . . . . . . . . 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 34981	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 6904	. . . . . . . . . . . . 13 ⊢ (1^st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49		fvex 6904	. . . . . . . . . . . . 13 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
50	48, 49	opth 5476	. . . . . . . . . . . 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 496	. . . . . . . . . . 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 6904	. . . . . . . . . . . 12 ⊢ ((mType‘𝑇)‘𝑣) ∈ V
54		vex 3477	. . . . . . . . . . . 12 ⊢ 𝑟 ∈ V
55	53, 54	op2nd 7988	. . . . . . . . . . 11 ⊢ (2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
56	55	fveq2i 6894	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑓)‘(2^nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
57	55	fveq2i 6894	. . . . . . . . . 10 ⊢ (((mRSubst‘𝑇)‘𝑔)‘(2^nd ‘⟨((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 7035	. . . . . . . 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 34970	. . . . . . . . . . 11 ⊢ (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅 ↑_m 𝑉)):(𝑅 ↑_m 𝑉)–1-1→(𝑅 ↑_m 𝑅))
61		f1fveq 7264	. . . . . . . . . . 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 579	. . . . . . . . . 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 6910	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑_m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64		fvres 6910	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅 ↑_m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
65	63, 64	eqeqan12d 2745	. . . . . . . . . . 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 6893	. . . . . 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 3153	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑔 ∈ (𝑅 ↑_m 𝑉))) → ((𝑆‘𝑓) = (𝑆‘𝑔) → ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
73		fvres 6910	. . . . . 6 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝑉) → ((𝑆 ↾ (𝑅 ↑_m 𝑉))‘𝑓) = (𝑆‘𝑓))
74		fvres 6910	. . . . . 6 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝑉) → ((𝑆 ↾ (𝑅 ↑_m 𝑉))‘𝑔) = (𝑆‘𝑔))
75	73, 74	eqeqan12d 2745	. . . . 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 6717	. . . . . . 7 ⊢ (𝑓:𝑉⟶𝑅 → 𝑓 Fn 𝑉)
78		ffn 6717	. . . . . . 7 ⊢ (𝑔:𝑉⟶𝑅 → 𝑔 Fn 𝑉)
79		eqfnfv 7032	. . . . . . 7 ⊢ ((𝑓 Fn 𝑉 ∧ 𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
80	77, 78, 79	syl2an 595	. . . . . 6 ⊢ ((𝑓:𝑉⟶𝑅 ∧ 𝑔:𝑉⟶𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣 ∈ 𝑉 (𝑓‘𝑣) = (𝑔‘𝑣)))
81	27, 43, 80	syl2an 595	. . . . 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 3199	. 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 7257	. 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 582	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 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3948 ⟨cop 4634 × cxp 5674 ↾ cres 5678 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 ↑_m cmap 8826 ↑_pm cpm 8827 mVRcmvar 34917 mTypecmty 34918 mTCcmtc 34920 mRExcmrex 34922 mExcmex 34923 mRSubstcmrsub 34926 mSubstcmsub 34927 mFScmfs 34932

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-gsum 17395 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-frmd 18772 df-mrex 34942 df-mex 34943 df-mrsub 34946 df-msub 34947 df-mfs 34952

This theorem is referenced by: msubff1o 35013

W3C validator