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Theorem funfvdm2f 5266
 Description: The value of a function. Version of funfvdm2 5265 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
Hypotheses
Ref Expression
funfvdm2f.1 𝑦𝐴
funfvdm2f.2 𝑦𝐹
Assertion
Ref Expression
funfvdm2f ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Proof of Theorem funfvdm2f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 funfvdm2 5265 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfvdm2f.1 . . . . 5 𝑦𝐴
3 funfvdm2f.2 . . . . 5 𝑦𝐹
4 nfcv 2194 . . . . 5 𝑦𝑤
52, 3, 4nfbr 3836 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1437 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 3796 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2176 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 3618 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9syl6eq 2104 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  {cab 2042  Ⅎwnfc 2181  ∪ cuni 3608   class class class wbr 3792  dom cdm 4373  Fun wfun 4924  ‘cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938 This theorem is referenced by: (None)
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