Theorem msubval 32767

Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

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)

Assertion

Ref	Expression
msubval	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)

Proof of Theorem msubval

Dummy variable 𝑒 is distinct from all other variables.

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 32766	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
7	6	3adant3 1128	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
8		simpr 487	. . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
9	8	fveq2d 6668	. . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (1st ‘𝑒) = (1st ‘𝑋))
10	8	fveq2d 6668	. . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (2nd ‘𝑒) = (2nd ‘𝑋))
11	10	fveq2d 6668	. . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ((𝑂‘𝐹)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))
12	9, 11	opeq12d 4804	. 2 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)
13		simp3 1134	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
14		opex 5348	. . 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 6769	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  ⟨cop 4566   ↦ cmpt 5138  ⟶wf 6345  ‘cfv 6349  1^st c1st 7681  2^nd c2nd 7682  mVRcmvar 32703  mRExcmrex 32708  mExcmex 32709  mRSubstcmrsub 32712  mSubstcmsub 32713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8403  df-msub 32733

This theorem is referenced by:  msubrsub  32768  msubty  32769  msubff1  32798