Theorem msubval 35723

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 35722	. . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
7	6	3adant3 1133	. 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 6838	. . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (1st ‘𝑒) = (1st ‘𝑋))
10	8	fveq2d 6838	. . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → (2nd ‘𝑒) = (2nd ‘𝑋))
11	10	fveq2d 6838	. . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ((𝑂‘𝐹)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))
12	9, 11	opeq12d 4825	. 2 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) ∧ 𝑒 = 𝑋) → ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)
13		simp3 1139	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸)
14		opex 5411	. . 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 6949	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ⟨cop 4574   ↦ cmpt 5167  ⟶wf 6488  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934  mVRcmvar 35659  mRExcmrex 35664  mExcmex 35665  mRSubstcmrsub 35668  mSubstcmsub 35669

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-pm 8769  df-msub 35689

This theorem is referenced by:  msubrsub  35724  msubty  35725  msubff1  35754