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Theorem fvmptf 5358
Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5343 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 = (𝑥𝐷𝐵)
Assertion
Ref Expression
fvmptf ((𝐴𝐷𝐶𝑉) → (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 2624 . . 3 (𝐶𝑉𝐶 ∈ V)
2 fvmptf.1 . . . 4 𝑥𝐴
3 fvmptf.2 . . . . . 6 𝑥𝐶
43nfel1 2235 . . . . 5 𝑥 𝐶 ∈ V
5 fvmptf.4 . . . . . . . 8 𝐹 = (𝑥𝐷𝐵)
6 nfmpt1 3906 . . . . . . . 8 𝑥(𝑥𝐷𝐵)
75, 6nfcxfr 2222 . . . . . . 7 𝑥𝐹
87, 2nffv 5278 . . . . . 6 𝑥(𝐹𝐴)
98, 3nfeq 2232 . . . . 5 𝑥(𝐹𝐴) = 𝐶
104, 9nfim 1507 . . . 4 𝑥(𝐶 ∈ V → (𝐹𝐴) = 𝐶)
11 fvmptf.3 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
1211eleq1d 2153 . . . . 5 (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V))
13 fveq2 5268 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
1413, 11eqeq12d 2099 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) = 𝐵 ↔ (𝐹𝐴) = 𝐶))
1512, 14imbi12d 232 . . . 4 (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹𝐴) = 𝐶)))
165fvmpt2 5349 . . . . 5 ((𝑥𝐷𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
1716ex 113 . . . 4 (𝑥𝐷 → (𝐵 ∈ V → (𝐹𝑥) = 𝐵))
182, 10, 15, 17vtoclgaf 2677 . . 3 (𝐴𝐷 → (𝐶 ∈ V → (𝐹𝐴) = 𝐶))
191, 18syl5 32 . 2 (𝐴𝐷 → (𝐶𝑉 → (𝐹𝐴) = 𝐶))
2019imp 122 1 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  wnfc 2212  Vcvv 2615  cmpt 3874  cfv 4981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-iota 4946  df-fun 4983  df-fv 4989
This theorem is referenced by:  isumrblem  10657  fisum  10667  isumss  10671
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