MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fndmeng Structured version   Visualization version   GIF version

Theorem fndmeng 9075
Description: A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
fndmeng ((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)

Proof of Theorem fndmeng
StepHypRef Expression
1 fnex 7237 . . 3 ((𝐹 Fn 𝐴𝐴𝐶) → 𝐹 ∈ V)
2 fnfun 6668 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
32adantr 480 . . 3 ((𝐹 Fn 𝐴𝐴𝐶) → Fun 𝐹)
4 fundmeng 9072 . . 3 ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹𝐹)
51, 3, 4syl2anc 584 . 2 ((𝐹 Fn 𝐴𝐴𝐶) → dom 𝐹𝐹)
6 fndm 6671 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76breq1d 5153 . . 3 (𝐹 Fn 𝐴 → (dom 𝐹𝐹𝐴𝐹))
87adantr 480 . 2 ((𝐹 Fn 𝐴𝐴𝐶) → (dom 𝐹𝐹𝐴𝐹))
95, 8mpbid 232 1 ((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  Vcvv 3480   class class class wbr 5143  dom cdm 5685  Fun wfun 6555   Fn wfn 6556  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-en 8986
This theorem is referenced by:  tskcard  10821  hashfn  14414
  Copyright terms: Public domain W3C validator