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Theorem fvmptf 5188
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5173 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 𝐷B)
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 2543 . . 3 (𝐶 𝑉𝐶 V)
2 fvmptf.1 . . . 4 xA
3 fvmptf.2 . . . . . 6 x𝐶
43nfel1 2170 . . . . 5 x 𝐶 V
5 fvmptf.4 . . . . . . . 8 𝐹 = (x 𝐷B)
6 nfmpt1 3824 . . . . . . . 8 x(x 𝐷B)
75, 6nfcxfr 2157 . . . . . . 7 x𝐹
87, 2nffv 5110 . . . . . 6 x(𝐹A)
98, 3nfeq 2167 . . . . 5 x(𝐹A) = 𝐶
104, 9nfim 1446 . . . 4 x(𝐶 V → (𝐹A) = 𝐶)
11 fvmptf.3 . . . . . 6 (x = AB = 𝐶)
1211eleq1d 2088 . . . . 5 (x = A → (B V ↔ 𝐶 V))
13 fveq2 5103 . . . . . 6 (x = A → (𝐹x) = (𝐹A))
1413, 11eqeq12d 2036 . . . . 5 (x = A → ((𝐹x) = B ↔ (𝐹A) = 𝐶))
1512, 14imbi12d 223 . . . 4 (x = A → ((B V → (𝐹x) = B) ↔ (𝐶 V → (𝐹A) = 𝐶)))
165fvmpt2 5179 . . . . 5 ((x 𝐷 B V) → (𝐹x) = B)
1716ex 108 . . . 4 (x 𝐷 → (B V → (𝐹x) = B))
182, 10, 15, 17vtoclgaf 2595 . . 3 (A 𝐷 → (𝐶 V → (𝐹A) = 𝐶))
191, 18syl5 28 . 2 (A 𝐷 → (𝐶 𝑉 → (𝐹A) = 𝐶))
2019imp 115 1 ((A 𝐷 𝐶 𝑉) → (𝐹A) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  wnfc 2147  Vcvv 2535  cmpt 3792  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fv 4837
This theorem is referenced by: (None)
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