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Theorem fndmeng 8019
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 6466 . . 3 ((𝐹 Fn 𝐴𝐴𝐶) → 𝐹 ∈ V)
2 fnfun 5976 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
32adantr 481 . . 3 ((𝐹 Fn 𝐴𝐴𝐶) → Fun 𝐹)
4 fundmeng 8016 . . 3 ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹𝐹)
51, 3, 4syl2anc 692 . 2 ((𝐹 Fn 𝐴𝐴𝐶) → dom 𝐹𝐹)
6 fndm 5978 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76breq1d 4654 . . 3 (𝐹 Fn 𝐴 → (dom 𝐹𝐹𝐴𝐹))
87adantr 481 . 2 ((𝐹 Fn 𝐴𝐴𝐶) → (dom 𝐹𝐹𝐴𝐹))
95, 8mpbid 222 1 ((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1988  Vcvv 3195   class class class wbr 4644  dom cdm 5104  Fun wfun 5870   Fn wfn 5871  cen 7937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-en 7941
This theorem is referenced by:  tskcard  9588  hashfn  13147
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