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Mirrors > Home > MPE Home > Th. List > Mathboxes > msubval | Structured version Visualization version GIF version |
Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
msubffval.v | ⊢ 𝑉 = (mVR‘𝑇) |
msubffval.r | ⊢ 𝑅 = (mREx‘𝑇) |
msubffval.s | ⊢ 𝑆 = (mSubst‘𝑇) |
msubffval.e | ⊢ 𝐸 = (mEx‘𝑇) |
msubffval.o | ⊢ 𝑂 = (mRSubst‘𝑇) |
Ref | Expression |
---|---|
msubval | ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msubffval.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | msubffval.r | . . . 4 ⊢ 𝑅 = (mREx‘𝑇) | |
3 | msubffval.s | . . . 4 ⊢ 𝑆 = (mSubst‘𝑇) | |
4 | msubffval.e | . . . 4 ⊢ 𝐸 = (mEx‘𝑇) | |
5 | msubffval.o | . . . 4 ⊢ 𝑂 = (mRSubst‘𝑇) | |
6 | 1, 2, 3, 4, 5 | msubfval 34118 | . . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉)) |
7 | 6 | 3adant3 1132 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉)) |
8 | simpr 485 | . . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋) | |
9 | 8 | fveq2d 6846 | . . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (1st ‘𝑒) = (1st ‘𝑋)) |
10 | 8 | fveq2d 6846 | . . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (2nd ‘𝑒) = (2nd ‘𝑋)) |
11 | 10 | fveq2d 6846 | . . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ((𝑂‘𝐹)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑋))) |
12 | 9, 11 | opeq12d 4838 | . 2 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉 = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉) |
13 | simp3 1138 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸) | |
14 | opex 5421 | . . 3 ⊢ 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉 ∈ V | |
15 | 14 | a1i 11 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉 ∈ V) |
16 | 7, 12, 13, 15 | fvmptd 6955 | 1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 〈cop 4592 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 1st c1st 7919 2nd c2nd 7920 mVRcmvar 34055 mRExcmrex 34060 mExcmex 34061 mRSubstcmrsub 34064 mSubstcmsub 34065 |
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 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3065 df-rex 3074 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-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 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-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-pm 8768 df-msub 34085 |
This theorem is referenced by: msubrsub 34120 msubty 34121 msubff1 34150 |
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