Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msubty Structured version   Visualization version   GIF version

Theorem msubty 34185
Description: The type of a substituted expression is the same as the original type. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msubffval.v 𝑉 = (mVRβ€˜π‘‡)
msubffval.r 𝑅 = (mRExβ€˜π‘‡)
msubffval.s 𝑆 = (mSubstβ€˜π‘‡)
msubffval.e 𝐸 = (mExβ€˜π‘‡)
Assertion
Ref Expression
msubty ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ (1st β€˜((π‘†β€˜πΉ)β€˜π‘‹)) = (1st β€˜π‘‹))

Proof of Theorem msubty
StepHypRef Expression
1 msubffval.v . . 3 𝑉 = (mVRβ€˜π‘‡)
2 msubffval.r . . 3 𝑅 = (mRExβ€˜π‘‡)
3 msubffval.s . . 3 𝑆 = (mSubstβ€˜π‘‡)
4 msubffval.e . . 3 𝐸 = (mExβ€˜π‘‡)
5 eqid 2733 . . 3 (mRSubstβ€˜π‘‡) = (mRSubstβ€˜π‘‡)
61, 2, 3, 4, 5msubval 34183 . 2 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ ((π‘†β€˜πΉ)β€˜π‘‹) = ⟨(1st β€˜π‘‹), (((mRSubstβ€˜π‘‡)β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩)
7 fvex 6859 . . 3 (1st β€˜π‘‹) ∈ V
8 fvex 6859 . . 3 (((mRSubstβ€˜π‘‡)β€˜πΉ)β€˜(2nd β€˜π‘‹)) ∈ V
97, 8op1std 7935 . 2 (((π‘†β€˜πΉ)β€˜π‘‹) = ⟨(1st β€˜π‘‹), (((mRSubstβ€˜π‘‡)β€˜πΉ)β€˜(2nd β€˜π‘‹))⟩ β†’ (1st β€˜((π‘†β€˜πΉ)β€˜π‘‹)) = (1st β€˜π‘‹))
106, 9syl 17 1 ((𝐹:π΄βŸΆπ‘… ∧ 𝐴 βŠ† 𝑉 ∧ 𝑋 ∈ 𝐸) β†’ (1st β€˜((π‘†β€˜πΉ)β€˜π‘‹)) = (1st β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3914  βŸ¨cop 4596  βŸΆwf 6496  β€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  mVRcmvar 34119  mRExcmrex 34124  mExcmex 34125  mRSubstcmrsub 34128  mSubstcmsub 34129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-pm 8774  df-msub 34149
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator