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Theorem fvmptd 5593
Description: Deduction version of fvmpt 5589. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fvmptd.1 (𝜑𝐹 = (𝑥𝐷𝐵))
fvmptd.2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
fvmptd.3 (𝜑𝐴𝐷)
fvmptd.4 (𝜑𝐶𝑉)
Assertion
Ref Expression
fvmptd (𝜑 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd
StepHypRef Expression
1 fvmptd.1 . . 3 (𝜑𝐹 = (𝑥𝐷𝐵))
21fveq1d 5513 . 2 (𝜑 → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
3 fvmptd.3 . . 3 (𝜑𝐴𝐷)
4 fvmptd.2 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
53, 4csbied 3103 . . . 4 (𝜑𝐴 / 𝑥𝐵 = 𝐶)
6 fvmptd.4 . . . 4 (𝜑𝐶𝑉)
75, 6eqeltrd 2254 . . 3 (𝜑𝐴 / 𝑥𝐵𝑉)
8 eqid 2177 . . . 4 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
98fvmpts 5590 . . 3 ((𝐴𝐷𝐴 / 𝑥𝐵𝑉) → ((𝑥𝐷𝐵)‘𝐴) = 𝐴 / 𝑥𝐵)
103, 7, 9syl2anc 411 . 2 (𝜑 → ((𝑥𝐷𝐵)‘𝐴) = 𝐴 / 𝑥𝐵)
112, 10, 53eqtrd 2214 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  csb 3057  cmpt 4061  cfv 5212
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220
This theorem is referenced by:  fvmptdv2  5601  rdgivallem  6376  1stinl  7067  2ndinl  7068  1stinr  7069  2ndinr  7070  updjudhcoinlf  7073  updjudhcoinrg  7074  cardcl  7174  caucvgsrlemfv  7778  caucvgsrlemoffval  7783  axcaucvglemval  7884  negiso  8898  infrenegsupex  9580  iseqf1olemfvp  10480  seq3f1olemqsum  10483  infxrnegsupex  11252  climcvg1nlem  11338  isumshft  11479  lmfval  13352  blfvalps  13545  cdivcncfap  13747  peano4nninf  14404  peano3nninf  14405  nninfsellemeq  14412  nninfsellemeqinf  14414
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