Theorem msubrsub 32773

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
msubrsub	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))

Proof of Theorem msubrsub

Step	Hyp	Ref	Expression
1		msubffval.v	. . 3 ⊢ 𝑉 = (mVR‘𝑇)
2		msubffval.r	. . 3 ⊢ 𝑅 = (mREx‘𝑇)
3		msubffval.s	. . 3 ⊢ 𝑆 = (mSubst‘𝑇)
4		msubffval.e	. . 3 ⊢ 𝐸 = (mEx‘𝑇)
5		msubffval.o	. . 3 ⊢ 𝑂 = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubval 32772	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩)
7		fvex 6683	. . 3 ⊢ (1st ‘𝑋) ∈ V
8		fvex 6683	. . 3 ⊢ ((𝑂‘𝐹)‘(2nd ‘𝑋)) ∈ V
9	7, 8	op2ndd 7700	. 2 ⊢ (((𝑆‘𝐹)‘𝑋) = ⟨(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))⟩ → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))
10	6, 9	syl 17	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋)))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  ⟨cop 4573  ⟶wf 6351  ‘cfv 6355  1^st c1st 7687  2^nd c2nd 7688  mVRcmvar 32708  mRExcmrex 32713  mExcmex 32714  mRSubstcmrsub 32717  mSubstcmsub 32718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-2nd 7690  df-pm 8409  df-msub 32738

This theorem is referenced by: (None)