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Theorem funfv2f 6429
Description: The value of a function. Version of funfv2 6428 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 6428 . 2 (Fun 𝐹 → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfv2f.1 . . . . 5 𝑦𝐴
3 funfv2f.2 . . . . 5 𝑦𝐹
4 nfcv 2902 . . . . 5 𝑦𝑤
52, 3, 4nfbr 4851 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1992 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 4808 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvab 2884 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 4597 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9syl6eq 2810 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  {cab 2746  wnfc 2889   cuni 4588   class class class wbr 4804  Fun wfun 6043  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by: (None)
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