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Theorem msubff1 31158
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 31132 . . 3 (𝑇 ∈ mFS → 𝑆:(𝑅pm 𝑉)⟶(𝐸𝑚 𝐸))
6 mapsspm 7835 . . . 4 (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉)
76a1i 11 . . 3 (𝑇 ∈ mFS → (𝑅𝑚 𝑉) ⊆ (𝑅pm 𝑉))
85, 7fssresd 6028 . 2 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)⟶(𝐸𝑚 𝐸))
9 eqid 2621 . . . . . . . . . . . . 13 (mRSubst‘𝑇) = (mRSubst‘𝑇)
101, 2, 9mrsubff 31114 . . . . . . . . . . . 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 3581 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅pm 𝑉))
1411, 13ffvelrnd 6316 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅))
15 elmapi 7823 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
16 ffn 6002 . . . . . . . . . 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 3581 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅pm 𝑉))
2011, 19ffvelrnd 6316 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅𝑚 𝑅))
21 elmapi 7823 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅𝑚 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅𝑅)
22 ffn 6002 . . . . . . . . . 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 6150 . . . . . . . . . . . 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 7823 . . . . . . . . . . . . . 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 3603 . . . . . . . . . . . . . 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 31153 . . . . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . 15 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
37 opelxpi 5108 . . . . . . . . . . . . . . 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 31104 . . . . . . . . . . . . . 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 31127 . . . . . . . . . . . . 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 7823 . . . . . . . . . . . . . 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 31127 . . . . . . . . . . . . 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 6158 . . . . . . . . . . . . 13 (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
50 fvex 6158 . . . . . . . . . . . . 13 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
5149, 50opth 4905 . . . . . . . . . . . 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 6158 . . . . . . . . . . . 12 ((mType‘𝑇)‘𝑣) ∈ V
55 vex 3189 . . . . . . . . . . . 12 𝑟 ∈ V
5654, 55op2nd 7122 . . . . . . . . . . 11 (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
5756fveq2i 6151 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
5856fveq2i 6151 . . . . . . . . . 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 6270 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
611, 2, 9mrsubff1 31116 . . . . . . . . . . 11 (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝑅𝑚 𝑅))
62 f1fveq 6473 . . . . . . . . . . 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 6164 . . . . . . . . . . . 12 (𝑓 ∈ (𝑅𝑚 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅𝑚 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
65 fvres 6164 . . . . . . . . . . . 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 6150 . . . . . 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 2963 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅𝑚 𝑉) ∧ 𝑔 ∈ (𝑅𝑚 𝑉))) → ((𝑆𝑓) = (𝑆𝑔) → ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
74 fvres 6164 . . . . . 6 (𝑓 ∈ (𝑅𝑚 𝑉) → ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = (𝑆𝑓))
75 fvres 6164 . . . . . 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 6002 . . . . . . 7 (𝑓:𝑉𝑅𝑓 Fn 𝑉)
79 ffn 6002 . . . . . . 7 (𝑔:𝑉𝑅𝑔 Fn 𝑉)
80 eqfnfv 6267 . . . . . . 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 2965 . 2 (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅𝑚 𝑉)∀𝑔 ∈ (𝑅𝑚 𝑉)(((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅𝑚 𝑉))‘𝑔) → 𝑓 = 𝑔))
86 dff13 6466 . 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 2907  wss 3555  cop 4154   × cxp 5072  cres 5076   Fn wfn 5842  wf 5843  1-1wf1 5844  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  𝑚 cmap 7802  pm cpm 7803  mVRcmvar 31063  mTypecmty 31064  mTCcmtc 31066  mRExcmrex 31068  mExcmex 31069  mRSubstcmrsub 31072  mSubstcmsub 31073  mFScmfs 31078
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-frmd 17307  df-mrex 31088  df-mex 31089  df-mrsub 31092  df-msub 31093  df-mfs 31098
This theorem is referenced by:  msubff1o  31159
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