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Theorem msubff1 34150
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 34124 . . 3 (𝑇 ∈ mFS → 𝑆:(𝑅pm 𝑉)⟶(𝐸m 𝐸))
6 mapsspm 8814 . . . 4 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
76a1i 11 . . 3 (𝑇 ∈ mFS → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
85, 7fssresd 6709 . 2 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)⟶(𝐸m 𝐸))
9 eqid 2736 . . . . . . . . . . . . 13 (mRSubst‘𝑇) = (mRSubst‘𝑇)
101, 2, 9mrsubff 34106 . . . . . . . . . . . 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 3942 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅pm 𝑉))
1411, 13ffvelcdmd 7036 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅m 𝑅))
15 elmapi 8787 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
16 ffn 6668 . . . . . . . . . 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 3942 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅pm 𝑉))
2011, 19ffvelcdmd 7036 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅m 𝑅))
21 elmapi 8787 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅𝑅)
22 ffn 6668 . . . . . . . . . 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 6844 . . . . . . . . . . . 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 8787 . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑉𝑉)
30 eqid 2736 . . . . . . . . . . . . . . . . . 18 (mTC‘𝑇) = (mTC‘𝑇)
31 eqid 2736 . . . . . . . . . . . . . . . . . 18 (mType‘𝑇) = (mType‘𝑇)
321, 30, 31mtyf2 34145 . . . . . . . . . . . . . . . . 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 7036 . . . . . . . . . . . . . . 15 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
36 opelxpi 5670 . . . . . . . . . . . . . . 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 34096 . . . . . . . . . . . . . 14 𝐸 = ((mTC‘𝑇) × 𝑅)
3937, 38eleqtrrdi 2849 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸)
401, 2, 3, 4, 9msubval 34119 . . . . . . . . . . . . 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 8787 . . . . . . . . . . . . . 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 34119 . . . . . . . . . . . . 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 2784 . . . . . . . . . . 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 6855 . . . . . . . . . . . . 13 (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49 fvex 6855 . . . . . . . . . . . . 13 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
5048, 49opth 5433 . . . . . . . . . . . 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 6855 . . . . . . . . . . . 12 ((mType‘𝑇)‘𝑣) ∈ V
54 vex 3449 . . . . . . . . . . . 12 𝑟 ∈ V
5553, 54op2nd 7930 . . . . . . . . . . 11 (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
5655fveq2i 6845 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
5755fveq2i 6845 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
5852, 56, 573eqtr3g 2799 . . . . . . . . 9 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (((mRSubst‘𝑇)‘𝑓)‘𝑟) = (((mRSubst‘𝑇)‘𝑔)‘𝑟))
5917, 23, 58eqfnfvd 6985 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
601, 2, 9mrsubff1 34108 . . . . . . . . . . 11 (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1→(𝑅m 𝑅))
61 f1fveq 7209 . . . . . . . . . . 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 6861 . . . . . . . . . . . 12 (𝑓 ∈ (𝑅m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64 fvres 6861 . . . . . . . . . . . 12 (𝑔 ∈ (𝑅m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
6563, 64eqeqan12d 2750 . . . . . . . . . . 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 6844 . . . . . 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 3151 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → ((𝑆𝑓) = (𝑆𝑔) → ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
73 fvres 6861 . . . . . 6 (𝑓 ∈ (𝑅m 𝑉) → ((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = (𝑆𝑓))
74 fvres 6861 . . . . . 6 (𝑔 ∈ (𝑅m 𝑉) → ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) = (𝑆𝑔))
7573, 74eqeqan12d 2750 . . . . 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 6668 . . . . . . 7 (𝑓:𝑉𝑅𝑓 Fn 𝑉)
78 ffn 6668 . . . . . . 7 (𝑔:𝑉𝑅𝑔 Fn 𝑉)
79 eqfnfv 6982 . . . . . . 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 3197 . 2 (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅m 𝑉)∀𝑔 ∈ (𝑅m 𝑉)(((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) → 𝑓 = 𝑔))
85 dff13 7202 . 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 3064  wss 3910  cop 4592   × cxp 5631  cres 5635   Fn wfn 6491  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  pm cpm 8766  mVRcmvar 34055  mTypecmty 34056  mTCcmtc 34058  mRExcmrex 34060  mExcmex 34061  mRSubstcmrsub 34064  mSubstcmsub 34065  mFScmfs 34070
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-frmd 18659  df-mrex 34080  df-mex 34081  df-mrsub 34084  df-msub 34085  df-mfs 34090
This theorem is referenced by:  msubff1o  34151
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