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Theorem msubff1 31214
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 31188 . . 3 (𝑇 ∈ mFS → 𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
6 mapsspm 7851 . . . 4 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
76a1i 11 . . 3 (𝑇 ∈ mFS → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
85, 7fssresd 6038 . 2 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)⟶(𝐸𝑚 𝐸))
9 eqid 2621 . . . . . . . . . . . . 13 (mRSubst‘𝑇) = (mRSubst‘𝑇)
101, 2, 9mrsubff 31170 . . . . . . . . . . . 12 (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
1110ad2antrr 761 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
12 simplrl 799 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅𝑚 𝑉))
136, 12sseldi 3586 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅pm 𝑉))
1411, 13ffvelrnd 6326 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅))
15 elmapi 7839 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
16 ffn 6012 . . . . . . . . . 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 800 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅𝑚 𝑉))
196, 18sseldi 3586 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅pm 𝑉))
2011, 19ffvelrnd 6326 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅𝑚 𝑅))
21 elmapi 7839 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅𝑅)
22 ffn 6012 . . . . . . . . . 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 800 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (𝑆𝑓) = (𝑆𝑔))
2524fveq1d 6160 . . . . . . . . . . . 12 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((𝑆𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ((𝑆𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))
2612adantr 481 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑓 ∈ (𝑅𝑚 𝑉))
27 elmapi 7839 . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . 14 𝑉𝑉
3029a1i 11 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑉𝑉)
31 eqid 2621 . . . . . . . . . . . . . . . . . 18 (mTC‘𝑇) = (mTC‘𝑇)
32 eqid 2621 . . . . . . . . . . . . . . . . . 18 (mType‘𝑇) = (mType‘𝑇)
331, 31, 32mtyf2 31209 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
3433ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
35 simplrl 799 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑣𝑉)
3634, 35ffvelrnd 6326 . . . . . . . . . . . . . . 15 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
37 opelxpi 5118 . . . . . . . . . . . . . . 15 ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3836, 37sylancom 700 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3931, 4, 2mexval 31160 . . . . . . . . . . . . . 14 𝐸 = ((mTC‘𝑇) × 𝑅)
4038, 39syl6eleqr 2709 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸)
411, 2, 3, 4, 9msubval 31183 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑔 ∈ (𝑅𝑚 𝑉))
44 elmapi 7839 . . . . . . . . . . . . . 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 31183 . . . . . . . . . . . . 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 1323 . . . . . . . . . . . 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 2663 . . . . . . . . . . 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 6168 . . . . . . . . . . . . 13 (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
50 fvex 6168 . . . . . . . . . . . . 13 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
5149, 50opth 4915 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 6168 . . . . . . . . . . . 12 ((mType‘𝑇)‘𝑣) ∈ V
55 vex 3193 . . . . . . . . . . . 12 𝑟 ∈ V
5654, 55op2nd 7137 . . . . . . . . . . 11 (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
5756fveq2i 6161 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
5856fveq2i 6161 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
5953, 57, 583eqtr3g 2678 . . . . . . . . 9 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (((mRSubst‘𝑇)‘𝑓)‘𝑟) = (((mRSubst‘𝑇)‘𝑔)‘𝑟))
6017, 23, 59eqfnfvd 6280 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
611, 2, 9mrsubff1 31172 . . . . . . . . . . 11 (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝑅𝑚 𝑅))
62 f1fveq 6484 . . . . . . . . . . 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 488 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
64 fvres 6174 . . . . . . . . . . . 12 (𝑓 ∈ (𝑅𝑚 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
65 fvres 6174 . . . . . . . . . . . 12 (𝑔 ∈ (𝑅𝑚 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
6664, 65eqeqan12d 2637 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → ((((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6766adantl 482 . . . . . . . . . 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 481 . . . . . . . 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 6160 . . . . . 6 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓𝑣) = (𝑔𝑣))
7271expr 642 . . . . 5 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ 𝑣𝑉) → ((𝑆𝑓) = (𝑆𝑔) → (𝑓𝑣) = (𝑔𝑣)))
7372ralrimdva 2965 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((𝑆𝑓) = (𝑆𝑔) → ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
74 fvres 6174 . . . . . 6 (𝑓 ∈ (𝑅𝑚 𝑉) → ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = (𝑆𝑓))
75 fvres 6174 . . . . . 6 (𝑔 ∈ (𝑅𝑚 𝑉) → ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) = (𝑆𝑔))
7674, 75eqeqan12d 2637 . . . . 5 ((𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
7776adantl 482 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → (((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
78 ffn 6012 . . . . . . 7 (𝑓:𝑉𝑅𝑓 Fn 𝑉)
79 ffn 6012 . . . . . . 7 (𝑔:𝑉𝑅𝑔 Fn 𝑉)
80 eqfnfv 6277 . . . . . . 7 ((𝑓 Fn 𝑉𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8178, 79, 80syl2an 494 . . . . . 6 ((𝑓:𝑉𝑅𝑔:𝑉𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8227, 44, 81syl2an 494 . . . . 5 ((𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8382adantl 482 . . . 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 2967 . 2 (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅𝑚 𝑉)∀𝑔 ∈ (𝑅𝑚 𝑉)(((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) → 𝑓 = 𝑔))
86 dff13 6477 . 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 697 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 384   = wceq 1480  wcel 1987  wral 2908  wss 3560  cop 4161   × cxp 5082  cres 5086   Fn wfn 5852  wf 5853  1-1wf1 5854  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  𝑚 cmap 7817  pm cpm 7818  mVRcmvar 31119  mTypecmty 31120  mTCcmtc 31122  mRExcmrex 31124  mExcmex 31125  mRSubstcmrsub 31128  mSubstcmsub 31129  mFScmfs 31134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-word 13254  df-concat 13256  df-s1 13257  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-0g 16042  df-gsum 16043  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-frmd 17326  df-mrex 31144  df-mex 31145  df-mrsub 31148  df-msub 31149  df-mfs 31154
This theorem is referenced by:  msubff1o  31215
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