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Theorem funfni 5183
 Description: Inference to convert a function and domain antecedent. (Contributed by set.mm contributors, 22-Apr-2004.)
Hypothesis
Ref Expression
funfni.1 ((Fun F B dom F) → φ)
Assertion
Ref Expression
funfni ((F Fn A B A) → φ)

Proof of Theorem funfni
StepHypRef Expression
1 fnfun 5181 . . 3 (F Fn A → Fun F)
21adantr 451 . 2 ((F Fn A B A) → Fun F)
3 fndm 5182 . . . 4 (F Fn A → dom F = A)
43eleq2d 2420 . . 3 (F Fn A → (B dom FB A))
54biimpar 471 . 2 ((F Fn A B A) → B dom F)
6 funfni.1 . 2 ((Fun F B dom F) → φ)
72, 5, 6syl2anc 642 1 ((F Fn A B A) → φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  dom cdm 4772  Fun wfun 4775   Fn wfn 4776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-fn 4790 This theorem is referenced by:  fneu  5187  elpreima  5407  fnopfv  5412  fnfvelrn  5414
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