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Theorem funfvdm2f 5565
Description: The value of a function. Version of funfvdm2 5564 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 5564 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfvdm2f.1 . . . . 5 𝑦𝐴
3 funfvdm2f.2 . . . . 5 𝑦𝐹
4 nfcv 2313 . . . . 5 𝑦𝑤
52, 3, 4nfbr 4036 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1522 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 3994 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2295 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 3807 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9eqtrdi 2220 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1349  wcel 2142  {cab 2157  wnfc 2300   cuni 3797   class class class wbr 3990  dom cdm 4612  Fun wfun 5194  cfv 5200
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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195
This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-sbc 2957  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-br 3991  df-opab 4052  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-fv 5208
This theorem is referenced by: (None)
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