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Theorem fvmptd2 5631
Description: Deduction version of fvmpt 5626 (where the definition of the mapping does not depend on the common antecedent  ph). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fvmptd2.1  |-  F  =  ( x  e.  D  |->  B )
fvmptd2.2  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
fvmptd2.3  |-  ( ph  ->  A  e.  D )
fvmptd2.4  |-  ( ph  ->  C  e.  V )
Assertion
Ref Expression
fvmptd2  |-  ( ph  ->  ( F `  A
)  =  C )
Distinct variable groups:    x, A    x, C    x, D    ph, x
Allowed substitution hints:    B( x)    F( x)    V( x)

Proof of Theorem fvmptd2
StepHypRef Expression
1 fvmptd2.1 . . 3  |-  F  =  ( x  e.  D  |->  B )
21a1i 9 . 2  |-  ( ph  ->  F  =  ( x  e.  D  |->  B ) )
3 fvmptd2.2 . 2  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
4 fvmptd2.3 . 2  |-  ( ph  ->  A  e.  D )
5 fvmptd2.4 . 2  |-  ( ph  ->  C  e.  V )
62, 3, 4, 5fvmptd 5630 1  |-  ( ph  ->  ( F `  A
)  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164    |-> cmpt 4090   ` cfv 5246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-iota 5207  df-fun 5248  df-fv 5254
This theorem is referenced by:  gausslemma2dlem2  15126  gausslemma2dlem3  15127
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