Theorem funfni 5223

Description: Inference to convert a function and domain antecedent. (Contributed by NM, 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

Step	Hyp	Ref	Expression
1		fnfun 5220	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	adantr 274	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → Fun 𝐹)
3		fndm 5222	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	eleq2d 2209	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
5	4	biimpar 295	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
6		funfni.1	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑)
7	2, 5, 6	syl2anc 408	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  dom cdm 4539  Fun wfun 5117   Fn wfn 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-fn 5126

This theorem is referenced by:  fneu  5227  fnbrfvb  5462  fvelrnb  5469  fvelimab  5477  fniinfv  5479  fvco2  5490  eqfnfv  5518  fndmdif  5525  fndmin  5527  elpreima  5539  fniniseg  5540  fniniseg2  5542  fnniniseg2  5543  rexsupp  5544  fnopfv  5550  fnfvelrn  5552  rexrn  5557  ralrn  5558  fsn2  5594  fnressn  5606  eufnfv  5648  rexima  5656  ralima  5657  fniunfv  5663  dff13  5669  foeqcnvco  5691  f1eqcocnv  5692  isocnv2  5713  isoini  5719  f1oiso  5727  fnovex  5804  suppssof1  5999  offveqb  6001  1stexg  6065  2ndexg  6066  smoiso  6199  rdgruledefgg  6272  rdgivallem  6278  frectfr  6297  frecrdg  6305  en1  6693  fnfi  6825  ordiso2  6920  slotex  11986