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Theorem fvmptf 5206
 Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5191 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 2560 . . 3 (𝐶 𝑉𝐶 V)
2 fvmptf.1 . . . 4 xA
3 fvmptf.2 . . . . . 6 x𝐶
43nfel1 2185 . . . . 5 x 𝐶 V
5 fvmptf.4 . . . . . . . 8 𝐹 = (x 𝐷B)
6 nfmpt1 3841 . . . . . . . 8 x(x 𝐷B)
75, 6nfcxfr 2172 . . . . . . 7 x𝐹
87, 2nffv 5128 . . . . . 6 x(𝐹A)
98, 3nfeq 2182 . . . . 5 x(𝐹A) = 𝐶
104, 9nfim 1461 . . . 4 x(𝐶 V → (𝐹A) = 𝐶)
11 fvmptf.3 . . . . . 6 (x = AB = 𝐶)
1211eleq1d 2103 . . . . 5 (x = A → (B V ↔ 𝐶 V))
13 fveq2 5121 . . . . . 6 (x = A → (𝐹x) = (𝐹A))
1413, 11eqeq12d 2051 . . . . 5 (x = A → ((𝐹x) = B ↔ (𝐹A) = 𝐶))
1512, 14imbi12d 223 . . . 4 (x = A → ((B V → (𝐹x) = B) ↔ (𝐶 V → (𝐹A) = 𝐶)))
165fvmpt2 5197 . . . . 5 ((x 𝐷 B V) → (𝐹x) = B)
1716ex 108 . . . 4 (x 𝐷 → (B V → (𝐹x) = B))
182, 10, 15, 17vtoclgaf 2612 . . 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 1242   ∈ wcel 1390  Ⅎwnfc 2162  Vcvv 2551   ↦ cmpt 3809  ‘cfv 4845 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853 This theorem is referenced by: (None)
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