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Theorem fvmptf 5722
 Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5698 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
fvmptf.1 xA
fvmptf.2 xC
fvmptf.3 (x = AB = C)
fvmptf.4 F = (x D B)
Assertion
Ref Expression
fvmptf ((A D C V) → (FA) = C)
Distinct variable group:   x,D
Allowed substitution hints:   A(x)   B(x)   C(x)   F(x)   V(x)

Proof of Theorem fvmptf
StepHypRef Expression
1 elex 2867 . . 3 (C VC V)
2 fvmptf.1 . . . 4 xA
3 fvmptf.2 . . . . . 6 xC
43nfel1 2499 . . . . 5 x C V
5 fvmptf.4 . . . . . . . 8 F = (x D B)
6 nfmpt1 5672 . . . . . . . 8 x(x D B)
75, 6nfcxfr 2486 . . . . . . 7 xF
87, 2nffv 5334 . . . . . 6 x(FA)
98, 3nfeq 2496 . . . . 5 x(FA) = C
104, 9nfim 1813 . . . 4 x(C V → (FA) = C)
11 fvmptf.3 . . . . . 6 (x = AB = C)
1211eleq1d 2419 . . . . 5 (x = A → (B V ↔ C V))
13 fveq2 5328 . . . . . 6 (x = A → (Fx) = (FA))
1413, 11eqeq12d 2367 . . . . 5 (x = A → ((Fx) = B ↔ (FA) = C))
1512, 14imbi12d 311 . . . 4 (x = A → ((B V → (Fx) = B) ↔ (C V → (FA) = C)))
165fvmpt2 5704 . . . . 5 ((x D B V) → (Fx) = B)
1716ex 423 . . . 4 (x D → (B V → (Fx) = B))
182, 10, 15, 17vtoclgaf 2919 . . 3 (A D → (C V → (FA) = C))
191, 18syl5 28 . 2 (A D → (C V → (FA) = C))
2019imp 418 1 ((A D C V) → (FA) = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859   ‘cfv 4781   ↦ cmpt 5651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652 This theorem is referenced by:  fvmptnf  5723
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