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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 30675 | . . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉)) |
7 | 6 | 3adant3 1074 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉)) |
8 | simpr 476 | . . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋) | |
9 | 8 | fveq2d 6107 | . . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (1st ‘𝑒) = (1st ‘𝑋)) |
10 | 8 | fveq2d 6107 | . . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (2nd ‘𝑒) = (2nd ‘𝑋)) |
11 | 10 | fveq2d 6107 | . . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ((𝑂‘𝐹)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑋))) |
12 | 9, 11 | opeq12d 4348 | . 2 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉 = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉) |
13 | simp3 1056 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸) | |
14 | opex 4859 | . . 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 6197 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 〈cop 4131 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 mVRcmvar 30612 mRExcmrex 30617 mExcmex 30618 mRSubstcmrsub 30621 mSubstcmsub 30622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-pm 7747 df-msub 30642 |
This theorem is referenced by: msubrsub 30677 msubty 30678 msubff1 30707 |
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