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Theorem msubff1 31760
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 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝐸𝑚 𝐸))

Proof of Theorem msubff1
Dummy variables 𝑓 𝑔 𝑟 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef 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)
51, 2, 3, 4msubff 31734 . . 3 (𝑇 ∈ mFS → 𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
6 mapsspm 8057 . . . 4 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
76a1i 11 . . 3 (𝑇 ∈ mFS → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
85, 7fssresd 6232 . 2 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)⟶(𝐸𝑚 𝐸))
9 eqid 2760 . . . . . . . . . . . . 13 (mRSubst‘𝑇) = (mRSubst‘𝑇)
101, 2, 9mrsubff 31716 . . . . . . . . . . . 12 (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
1110ad2antrr 764 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
12 simplrl 819 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅𝑚 𝑉))
136, 12sseldi 3742 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅pm 𝑉))
1411, 13ffvelrnd 6523 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅))
15 elmapi 8045 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
16 ffn 6206 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓):𝑅𝑅 → ((mRSubst‘𝑇)‘𝑓) Fn 𝑅)
1714, 15, 163syl 18 . . . . . . . . 9 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) Fn 𝑅)
18 simplrr 820 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅𝑚 𝑉))
196, 18sseldi 3742 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅pm 𝑉))
2011, 19ffvelrnd 6523 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅𝑚 𝑅))
21 elmapi 8045 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅𝑅)
22 ffn 6206 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔):𝑅𝑅 → ((mRSubst‘𝑇)‘𝑔) Fn 𝑅)
2320, 21, 223syl 18 . . . . . . . . 9 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) Fn 𝑅)
24 simplrr 820 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (𝑆𝑓) = (𝑆𝑔))
2524fveq1d 6354 . . . . . . . . . . . 12 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((𝑆𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ((𝑆𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))
2612adantr 472 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑓 ∈ (𝑅𝑚 𝑉))
27 elmapi 8045 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅𝑚 𝑉) → 𝑓:𝑉𝑅)
2826, 27syl 17 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑓:𝑉𝑅)
29 ssid 3765 . . . . . . . . . . . . . 14 𝑉𝑉
3029a1i 11 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑉𝑉)
31 eqid 2760 . . . . . . . . . . . . . . . . . 18 (mTC‘𝑇) = (mTC‘𝑇)
32 eqid 2760 . . . . . . . . . . . . . . . . . 18 (mType‘𝑇) = (mType‘𝑇)
331, 31, 32mtyf2 31755 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
3433ad3antrrr 768 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
35 simplrl 819 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑣𝑉)
3634, 35ffvelrnd 6523 . . . . . . . . . . . . . . 15 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
37 opelxpi 5305 . . . . . . . . . . . . . . 15 ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3836, 37sylancom 704 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3931, 4, 2mexval 31706 . . . . . . . . . . . . . 14 𝐸 = ((mTC‘𝑇) × 𝑅)
4038, 39syl6eleqr 2850 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸)
411, 2, 3, 4, 9msubval 31729 . . . . . . . . . . . . 13 ((𝑓:𝑉𝑅𝑉𝑉 ∧ ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸) → ((𝑆𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
4228, 30, 40, 41syl3anc 1477 . . . . . . . . . . . 12 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((𝑆𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
4318adantr 472 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑔 ∈ (𝑅𝑚 𝑉))
44 elmapi 8045 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝑅𝑚 𝑉) → 𝑔:𝑉𝑅)
4543, 44syl 17 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑔:𝑉𝑅)
461, 2, 3, 4, 9msubval 31729 . . . . . . . . . . . . 13 ((𝑔:𝑉𝑅𝑉𝑉 ∧ ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸) → ((𝑆𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
4745, 30, 40, 46syl3anc 1477 . . . . . . . . . . . 12 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((𝑆𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ⟨(1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩), (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))⟩)
4825, 42, 473eqtr3d 2802 . . . . . . . . . . 11 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨(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⟩))⟩)
49 fvex 6362 . . . . . . . . . . . . 13 (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
50 fvex 6362 . . . . . . . . . . . . 13 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
5149, 50opth 5093 . . . . . . . . . . . 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⟩))))
5251simprbi 483 . . . . . . . . . . 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⟩)))
5348, 52syl 17 . . . . . . . . . 10 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)))
54 fvex 6362 . . . . . . . . . . . 12 ((mType‘𝑇)‘𝑣) ∈ V
55 vex 3343 . . . . . . . . . . . 12 𝑟 ∈ V
5654, 55op2nd 7342 . . . . . . . . . . 11 (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
5756fveq2i 6355 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
5856fveq2i 6355 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
5953, 57, 583eqtr3g 2817 . . . . . . . . 9 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (((mRSubst‘𝑇)‘𝑓)‘𝑟) = (((mRSubst‘𝑇)‘𝑔)‘𝑟))
6017, 23, 59eqfnfvd 6477 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
611, 2, 9mrsubff1 31718 . . . . . . . . . . 11 (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝑅𝑚 𝑅))
62 f1fveq 6682 . . . . . . . . . . 11 ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝑅𝑚 𝑅) ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
6361, 62sylan 489 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
64 fvres 6368 . . . . . . . . . . . 12 (𝑓 ∈ (𝑅𝑚 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
65 fvres 6368 . . . . . . . . . . . 12 (𝑔 ∈ (𝑅𝑚 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
6664, 65eqeqan12d 2776 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6766adantl 473 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6863, 67bitr3d 270 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6968adantr 472 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
7060, 69mpbird 247 . . . . . . 7 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 = 𝑔)
7170fveq1d 6354 . . . . . 6 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓𝑣) = (𝑔𝑣))
7271expr 644 . . . . 5 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ 𝑣𝑉) → ((𝑆𝑓) = (𝑆𝑔) → (𝑓𝑣) = (𝑔𝑣)))
7372ralrimdva 3107 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((𝑆𝑓) = (𝑆𝑔) → ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
74 fvres 6368 . . . . . 6 (𝑓 ∈ (𝑅𝑚 𝑉) → ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = (𝑆𝑓))
75 fvres 6368 . . . . . 6 (𝑔 ∈ (𝑅𝑚 𝑉) → ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) = (𝑆𝑔))
7674, 75eqeqan12d 2776 . . . . 5 ((𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
7776adantl 473 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → (((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
78 ffn 6206 . . . . . . 7 (𝑓:𝑉𝑅𝑓 Fn 𝑉)
79 ffn 6206 . . . . . . 7 (𝑔:𝑉𝑅𝑔 Fn 𝑉)
80 eqfnfv 6474 . . . . . . 7 ((𝑓 Fn 𝑉𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8178, 79, 80syl2an 495 . . . . . 6 ((𝑓:𝑉𝑅𝑔:𝑉𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8227, 44, 81syl2an 495 . . . . 5 ((𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8382adantl 473 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8473, 77, 833imtr4d 283 . . 3 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → (((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) → 𝑓 = 𝑔))
8584ralrimivva 3109 . 2 (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅𝑚 𝑉)∀𝑔 ∈ (𝑅𝑚 𝑉)(((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) → 𝑓 = 𝑔))
86 dff13 6675 . 2 ((𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝐸𝑚 𝐸) ↔ ((𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)⟶(𝐸𝑚 𝐸) ∧ ∀𝑓 ∈ (𝑅𝑚 𝑉)∀𝑔 ∈ (𝑅𝑚 𝑉)(((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) → 𝑓 = 𝑔)))
878, 85, 86sylanbrc 701 1 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝐸𝑚 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wss 3715  cop 4327   × cxp 5264  cres 5268   Fn wfn 6044  wf 6045  1-1wf1 6046  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  𝑚 cmap 8023  pm cpm 8024  mVRcmvar 31665  mTypecmty 31666  mTCcmtc 31668  mRExcmrex 31670  mExcmex 31671  mRSubstcmrsub 31674  mSubstcmsub 31675  mFScmfs 31680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-word 13485  df-concat 13487  df-s1 13488  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-gsum 16305  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-frmd 17587  df-mrex 31690  df-mex 31691  df-mrsub 31694  df-msub 31695  df-mfs 31700
This theorem is referenced by:  msubff1o  31761
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