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Theorem funfv2f 6731
 Description: The value of a function. Version of funfv2 6730 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 6730 . 2 (Fun 𝐹 → (𝐹𝐴) = {𝑤𝐴𝐹𝑤})
2 funfv2f.1 . . . . 5 𝑦𝐴
3 funfv2f.2 . . . . 5 𝑦𝐹
4 nfcv 2958 . . . . 5 𝑦𝑤
52, 3, 4nfbr 5080 . . . 4 𝑦 𝐴𝐹𝑤
6 nfv 1915 . . . 4 𝑤 𝐴𝐹𝑦
7 breq2 5037 . . . 4 (𝑤 = 𝑦 → (𝐴𝐹𝑤𝐴𝐹𝑦))
85, 6, 7cbvabw 2870 . . 3 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
98unieqi 4816 . 2 {𝑤𝐴𝐹𝑤} = {𝑦𝐴𝐹𝑦}
101, 9eqtrdi 2852 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  {cab 2779  Ⅎwnfc 2939  ∪ cuni 4803   class class class wbr 5033  Fun wfun 6322  ‘cfv 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336 This theorem is referenced by: (None)
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