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Theorem funfv2f 6426
Description: The value of a function. Version of funfv2 6425 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
Hypotheses
Ref Expression
funfv2f.1 𝑦𝐴
funfv2f.2 𝑦𝐹
Assertion
Ref Expression
funfv2f (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})

Proof of Theorem funfv2f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 funfv2 6425 . 2 (Fun 𝐹 → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfv2f.1 . . . . 5 𝑦𝐴
3 funfv2f.2 . . . . 5 𝑦𝐹
4 nfcv 2916 . . . . 5 𝑦𝑤
52, 3, 4nfbr 4844 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1998 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 4801 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2898 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 4594 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9syl6eq 2824 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1634  {cab 2760  wnfc 2903   cuni 4585   class class class wbr 4797  Fun wfun 6036  cfv 6042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-fv 6050
This theorem is referenced by: (None)
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