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Theorem fvmptdv 5612
Description: Alternate deduction version of fvmpt 5602, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdf.1  |-  ( ph  ->  A  e.  D )
fvmptdf.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
fvmptdf.3  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
Assertion
Ref Expression
fvmptdv  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Distinct variable groups:    x, A    x, D    ph, x    x, F    ps, x
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem fvmptdv
StepHypRef Expression
1 fvmptdf.1 . 2  |-  ( ph  ->  A  e.  D )
2 fvmptdf.2 . 2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
3 fvmptdf.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  (
( F `  A
)  =  B  ->  ps ) )
4 nfcv 2419 . 2  |-  F/_ x F
5 nfv 1605 . 2  |-  F/ x ps
61, 2, 3, 4, 5fvmptdf 5611 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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