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Theorem funfvdm2f 5535
Description: The value of a function. Version of funfvdm2 5534 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 5534 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfvdm2f.1 . . . . 5 𝑦𝐴
3 funfvdm2f.2 . . . . 5 𝑦𝐹
4 nfcv 2299 . . . . 5 𝑦𝑤
52, 3, 4nfbr 4012 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1508 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 3971 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2281 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 3784 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9eqtrdi 2206 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  {cab 2143  wnfc 2286   cuni 3774   class class class wbr 3967  dom cdm 4588  Fun wfun 5166  cfv 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180
This theorem is referenced by: (None)
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