Theorem msubval 34971

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 34970	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
7	6	3adant3 1129	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
8		simpr 484	. . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
9	8	fveq2d 6885	. . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (1st ‘𝑒) = (1st ‘𝑋))
10	8	fveq2d 6885	. . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (2nd ‘𝑒) = (2nd ‘𝑋))
11	10	fveq2d 6885	. . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ((𝑂‘𝐹)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))
12	9, 11	opeq12d 4873	. 2 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)
13		simp3 1135	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
14		opex 5454	. . 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 6995	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3940  ⟨cop 4626   ↦ cmpt 5221  ⟶wf 6529  ‘cfv 6533  1^st c1st 7966  2^nd c2nd 7967  mVRcmvar 34907  mRExcmrex 34912  mExcmex 34913  mRSubstcmrsub 34916  mSubstcmsub 34917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-pm 8818  df-msub 34937

This theorem is referenced by:  msubrsub  34972  msubty  34973  msubff1  35002