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Theorem msubff1 33034
 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 33008 . . 3 (𝑇 ∈ mFS → 𝑆:(𝑅pm 𝑉)⟶(𝐸m 𝐸))
6 mapsspm 8458 . . . 4 (𝑅m 𝑉) ⊆ (𝑅pm 𝑉)
76a1i 11 . . 3 (𝑇 ∈ mFS → (𝑅m 𝑉) ⊆ (𝑅pm 𝑉))
85, 7fssresd 6530 . 2 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)⟶(𝐸m 𝐸))
9 eqid 2758 . . . . . . . . . . . . 13 (mRSubst‘𝑇) = (mRSubst‘𝑇)
101, 2, 9mrsubff 32990 . . . . . . . . . . . 12 (𝑇 ∈ mFS → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
1110ad2antrr 725 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (mRSubst‘𝑇):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
12 simplrl 776 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅m 𝑉))
136, 12sseldi 3890 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 ∈ (𝑅pm 𝑉))
1411, 13ffvelrnd 6843 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅m 𝑅))
15 elmapi 8438 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅𝑅)
16 ffn 6498 . . . . . . . . . 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 777 . . . . . . . . . . . 12 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅m 𝑉))
196, 18sseldi 3890 . . . . . . . . . . 11 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑔 ∈ (𝑅pm 𝑉))
2011, 19ffvelrnd 6843 . . . . . . . . . 10 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑔) ∈ (𝑅m 𝑅))
21 elmapi 8438 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔) ∈ (𝑅m 𝑅) → ((mRSubst‘𝑇)‘𝑔):𝑅𝑅)
22 ffn 6498 . . . . . . . . . 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 777 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (𝑆𝑓) = (𝑆𝑔))
2524fveq1d 6660 . . . . . . . . . . . 12 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((𝑆𝑓)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = ((𝑆𝑔)‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩))
2612adantr 484 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑓 ∈ (𝑅m 𝑉))
27 elmapi 8438 . . . . . . . . . . . . . 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 3915 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑉𝑉)
30 eqid 2758 . . . . . . . . . . . . . . . . . 18 (mTC‘𝑇) = (mTC‘𝑇)
31 eqid 2758 . . . . . . . . . . . . . . . . . 18 (mType‘𝑇) = (mType‘𝑇)
321, 30, 31mtyf2 33029 . . . . . . . . . . . . . . . . 17 (𝑇 ∈ mFS → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
3332ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (mType‘𝑇):𝑉⟶(mTC‘𝑇))
34 simplrl 776 . . . . . . . . . . . . . . . 16 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑣𝑉)
3533, 34ffvelrnd 6843 . . . . . . . . . . . . . . 15 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇))
36 opelxpi 5561 . . . . . . . . . . . . . . 15 ((((mType‘𝑇)‘𝑣) ∈ (mTC‘𝑇) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3735, 36sylancom 591 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ ((mTC‘𝑇) × 𝑅))
3830, 4, 2mexval 32980 . . . . . . . . . . . . . 14 𝐸 = ((mTC‘𝑇) × 𝑅)
3937, 38eleqtrrdi 2863 . . . . . . . . . . . . 13 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → ⟨((mType‘𝑇)‘𝑣), 𝑟⟩ ∈ 𝐸)
401, 2, 3, 4, 9msubval 33003 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → 𝑔 ∈ (𝑅m 𝑉))
43 elmapi 8438 . . . . . . . . . . . . . 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 33003 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 2801 . . . . . . . . . . 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 6671 . . . . . . . . . . . . 13 (1st ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) ∈ V
49 fvex 6671 . . . . . . . . . . . . 13 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) ∈ V
5048, 49opth 5336 . . . . . . . . . . . 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 500 . . . . . . . . . . 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 6671 . . . . . . . . . . . 12 ((mType‘𝑇)‘𝑣) ∈ V
54 vex 3413 . . . . . . . . . . . 12 𝑟 ∈ V
5553, 54op2nd 7702 . . . . . . . . . . 11 (2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩) = 𝑟
5655fveq2i 6661 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑓)‘𝑟)
5755fveq2i 6661 . . . . . . . . . 10 (((mRSubst‘𝑇)‘𝑔)‘(2nd ‘⟨((mType‘𝑇)‘𝑣), 𝑟⟩)) = (((mRSubst‘𝑇)‘𝑔)‘𝑟)
5852, 56, 573eqtr3g 2816 . . . . . . . . 9 ((((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) ∧ 𝑟𝑅) → (((mRSubst‘𝑇)‘𝑓)‘𝑟) = (((mRSubst‘𝑇)‘𝑔)‘𝑟))
5917, 23, 58eqfnfvd 6796 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔))
601, 2, 9mrsubff1 32992 . . . . . . . . . . 11 (𝑇 ∈ mFS → ((mRSubst‘𝑇) ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1→(𝑅m 𝑅))
61 f1fveq 7012 . . . . . . . . . . 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 583 . . . . . . . . . 10 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → ((((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) ↔ 𝑓 = 𝑔))
63 fvres 6677 . . . . . . . . . . . 12 (𝑓 ∈ (𝑅m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = ((mRSubst‘𝑇)‘𝑓))
64 fvres 6677 . . . . . . . . . . . 12 (𝑔 ∈ (𝑅m 𝑉) → (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) = ((mRSubst‘𝑇)‘𝑔))
6563, 64eqeqan12d 2775 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → ((((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑓) = (((mRSubst‘𝑇) ↾ (𝑅m 𝑉))‘𝑔) ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6665adantl 485 . . . . . . . . . 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 284 . . . . . . . . 9 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6867adantr 484 . . . . . . . 8 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓 = 𝑔 ↔ ((mRSubst‘𝑇)‘𝑓) = ((mRSubst‘𝑇)‘𝑔)))
6959, 68mpbird 260 . . . . . . 7 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → 𝑓 = 𝑔)
7069fveq1d 6660 . . . . . 6 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ (𝑣𝑉 ∧ (𝑆𝑓) = (𝑆𝑔))) → (𝑓𝑣) = (𝑔𝑣))
7170expr 460 . . . . 5 (((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) ∧ 𝑣𝑉) → ((𝑆𝑓) = (𝑆𝑔) → (𝑓𝑣) = (𝑔𝑣)))
7271ralrimdva 3118 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → ((𝑆𝑓) = (𝑆𝑔) → ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
73 fvres 6677 . . . . . 6 (𝑓 ∈ (𝑅m 𝑉) → ((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = (𝑆𝑓))
74 fvres 6677 . . . . . 6 (𝑔 ∈ (𝑅m 𝑉) → ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) = (𝑆𝑔))
7573, 74eqeqan12d 2775 . . . . 5 ((𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
7675adantl 485 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) ↔ (𝑆𝑓) = (𝑆𝑔)))
77 ffn 6498 . . . . . . 7 (𝑓:𝑉𝑅𝑓 Fn 𝑉)
78 ffn 6498 . . . . . . 7 (𝑔:𝑉𝑅𝑔 Fn 𝑉)
79 eqfnfv 6793 . . . . . . 7 ((𝑓 Fn 𝑉𝑔 Fn 𝑉) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8077, 78, 79syl2an 598 . . . . . 6 ((𝑓:𝑉𝑅𝑔:𝑉𝑅) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8127, 43, 80syl2an 598 . . . . 5 ((𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉)) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8281adantl 485 . . . 4 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (𝑓 = 𝑔 ↔ ∀𝑣𝑉 (𝑓𝑣) = (𝑔𝑣)))
8372, 76, 823imtr4d 297 . . 3 ((𝑇 ∈ mFS ∧ (𝑓 ∈ (𝑅m 𝑉) ∧ 𝑔 ∈ (𝑅m 𝑉))) → (((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) → 𝑓 = 𝑔))
8483ralrimivva 3120 . 2 (𝑇 ∈ mFS → ∀𝑓 ∈ (𝑅m 𝑉)∀𝑔 ∈ (𝑅m 𝑉)(((𝑆 ↾ (𝑅m 𝑉))‘𝑓) = ((𝑆 ↾ (𝑅m 𝑉))‘𝑔) → 𝑓 = 𝑔))
85 dff13 7005 . 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 586 1 (𝑇 ∈ mFS → (𝑆 ↾ (𝑅m 𝑉)):(𝑅m 𝑉)–1-1→(𝐸m 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3858  ⟨cop 4528   × cxp 5522   ↾ cres 5526   Fn wfn 6330  ⟶wf 6331  –1-1→wf1 6332  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692   ↑m cmap 8416   ↑pm cpm 8417  mVRcmvar 32939  mTypecmty 32940  mTCcmtc 32942  mRExcmrex 32944  mExcmex 32945  mRSubstcmrsub 32948  mSubstcmsub 32949  mFScmfs 32954 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-seq 13419  df-hash 13741  df-word 13914  df-concat 13970  df-s1 13997  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-0g 16773  df-gsum 16774  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-submnd 18023  df-frmd 18080  df-mrex 32964  df-mex 32965  df-mrsub 32968  df-msub 32969  df-mfs 32974 This theorem is referenced by:  msubff1o  33035
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