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Theorem msubff1 34378
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 34352 . . 3 (𝑇 ∈ mFS → 𝑆:(𝑅pm 𝑉)⟶(𝐸m 𝐸))
6 mapsspm 8853 . . . 4 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
76a1i 11 . . 3 (𝑇 ∈ mFS → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
85, 7fssresd 6745 . 2 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)⟶(𝐸m 𝐸))
9 eqid 2731 . . . . . . . . . . . . 13 (mRSubst‘𝑇) = (mRSubst‘𝑇)
101, 2, 9mrsubff 34334 . . . . . . . . . . . 12 (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
1110ad2antrr 724 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
12 simplrl 775 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅m 𝑉))
136, 12sselid 3976 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅pm 𝑉))
1411, 13ffvelcdmd 7072 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅m 𝑅))
15 elmapi 8826 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
16 ffn 6704 . . . . . . . . . 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 776 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅m 𝑉))
196, 18sselid 3976 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅pm 𝑉))
2011, 19ffvelcdmd 7072 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅m 𝑅))
21 elmapi 8826 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅𝑅)
22 ffn 6704 . . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (𝑆𝑓) = (𝑆𝑔))
2524fveq1d 6880 . . . . . . . . . . . 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 8826 . . . . . . . . . . . . . 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 ssidd 4001 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑉𝑉)
30 eqid 2731 . . . . . . . . . . . . . . . . . 18 (mTC‘𝑇) = (mTC‘𝑇)
31 eqid 2731 . . . . . . . . . . . . . . . . . 18 (mType‘𝑇) = (mType‘𝑇)
321, 30, 31mtyf2 34373 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
3332ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
34 simplrl 775 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑣𝑉)
3533, 34ffvelcdmd 7072 . . . . . . . . . . . . . . 15 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
36 opelxpi 5706 . . . . . . . . . . . . . . 15 ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3735, 36sylancom 588 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3830, 4, 2mexval 34324 . . . . . . . . . . . . . 14 𝐸 = ((mTC‘𝑇) × 𝑅)
3937, 38eleqtrrdi 2843 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸)
401, 2, 3, 4, 9msubval 34347 . . . . . . . . . . . . 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⟩))⟩)
4128, 29, 39, 40syl3anc 1371 . . . . . . . . . . . 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⟩))⟩)
4218adantr 481 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑔 ∈ (𝑅m 𝑉))
43 elmapi 8826 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝑅m 𝑉) → 𝑔:𝑉𝑅)
4442, 43syl 17 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑔:𝑉𝑅)
451, 2, 3, 4, 9msubval 34347 . . . . . . . . . . . . 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⟩))⟩)
4644, 29, 39, 45syl3anc 1371 . . . . . . . . . . . 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⟩))⟩)
4725, 41, 463eqtr3d 2779 . . . . . . . . . . 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⟩))⟩)
48 fvex 6891 . . . . . . . . . . . . 13 (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49 fvex 6891 . . . . . . . . . . . . 13 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
5048, 49opth 5469 . . . . . . . . . . . 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⟩))))
5150simprbi 497 . . . . . . . . . . 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⟩)))
5247, 51syl 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⟩)))
53 fvex 6891 . . . . . . . . . . . 12 ((mType‘𝑇)‘𝑣) ∈ V
54 vex 3477 . . . . . . . . . . . 12 𝑟 ∈ V
5553, 54op2nd 7966 . . . . . . . . . . 11 (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
5655fveq2i 6881 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
5755fveq2i 6881 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
5852, 56, 573eqtr3g 2794 . . . . . . . . 9 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (((mRSubst‘𝑇)‘𝑓)‘𝑟) = (((mRSubst‘𝑇)‘𝑔)‘𝑟))
5917, 23, 58eqfnfvd 7021 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
601, 2, 9mrsubff1 34336 . . . . . . . . . . 11 (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1→(𝑅m 𝑅))
61 f1fveq 7245 . . . . . . . . . . 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))
6260, 61sylan 580 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
63 fvres 6897 . . . . . . . . . . . 12 (𝑓 ∈ (𝑅m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64 fvres 6897 . . . . . . . . . . . 12 (𝑔 ∈ (𝑅m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
6563, 64eqeqan12d 2745 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → ((((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6665adantl 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)))
6762, 66bitr3d 280 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6867adantr 481 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6959, 68mpbird 256 . . . . . . 7 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 = 𝑔)
7069fveq1d 6880 . . . . . 6 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓𝑣) = (𝑔𝑣))
7170expr 457 . . . . 5 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ 𝑣𝑉) → ((𝑆𝑓) = (𝑆𝑔) → (𝑓𝑣) = (𝑔𝑣)))
7271ralrimdva 3153 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → ((𝑆𝑓) = (𝑆𝑔) → ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
73 fvres 6897 . . . . . 6 (𝑓 ∈ (𝑅m 𝑉) → ((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = (𝑆𝑓))
74 fvres 6897 . . . . . 6 (𝑔 ∈ (𝑅m 𝑉) → ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) = (𝑆𝑔))
7573, 74eqeqan12d 2745 . . . . 5 ((𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
7675adantl 482 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
77 ffn 6704 . . . . . . 7 (𝑓:𝑉𝑅𝑓 Fn 𝑉)
78 ffn 6704 . . . . . . 7 (𝑔:𝑉𝑅𝑔 Fn 𝑉)
79 eqfnfv 7018 . . . . . . 7 ((𝑓 Fn 𝑉𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8077, 78, 79syl2an 596 . . . . . 6 ((𝑓:𝑉𝑅𝑔:𝑉𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8127, 43, 80syl2an 596 . . . . 5 ((𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8281adantl 482 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8372, 76, 823imtr4d 293 . . 3 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) → 𝑓 = 𝑔))
8483ralrimivva 3199 . 2 (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅m 𝑉)∀𝑔 ∈ (𝑅m 𝑉)(((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) → 𝑓 = 𝑔))
85 dff13 7238 . 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)))
868, 84, 85sylanbrc 583 1 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1→(𝐸m 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  wss 3944  cop 4628   × cxp 5667  cres 5671   Fn wfn 6527  wf 6528  1-1wf1 6529  cfv 6532  (class class class)co 7393  1st c1st 7955  2nd c2nd 7956  m cmap 8803  pm cpm 8804  mVRcmvar 34283  mTypecmty 34284  mTCcmtc 34286  mRExcmrex 34288  mExcmex 34289  mRSubstcmrsub 34292  mSubstcmsub 34293  mFScmfs 34298
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
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 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 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-pm 8806  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-fzo 13610  df-seq 13949  df-hash 14273  df-word 14447  df-concat 14503  df-s1 14528  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-0g 17369  df-gsum 17370  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-submnd 18648  df-frmd 18705  df-mrex 34308  df-mex 34309  df-mrsub 34312  df-msub 34313  df-mfs 34318
This theorem is referenced by:  msubff1o  34379
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