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Theorem funfv2f 6306
 Description: The value of a function. Version of funfv2 6305 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 6305 . 2 (Fun 𝐹 → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfv2f.1 . . . . 5 𝑦𝐴
3 funfv2f.2 . . . . 5 𝑦𝐹
4 nfcv 2793 . . . . 5 𝑦𝑤
52, 3, 4nfbr 4732 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1883 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 4689 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2775 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 4477 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9syl6eq 2701 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523  {cab 2637  Ⅎwnfc 2780  ∪ cuni 4468   class class class wbr 4685  Fun wfun 5920  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by: (None)
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