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Theorem fndmeng 8384
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 6806 . . 3 ((𝐹 Fn 𝐴𝐴𝐶) → 𝐹 ∈ V)
2 fnfun 6286 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
32adantr 473 . . 3 ((𝐹 Fn 𝐴𝐴𝐶) → Fun 𝐹)
4 fundmeng 8381 . . 3 ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹𝐹)
51, 3, 4syl2anc 576 . 2 ((𝐹 Fn 𝐴𝐴𝐶) → dom 𝐹𝐹)
6 fndm 6288 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76breq1d 4939 . . 3 (𝐹 Fn 𝐴 → (dom 𝐹𝐹𝐴𝐹))
87adantr 473 . 2 ((𝐹 Fn 𝐴𝐴𝐶) → (dom 𝐹𝐹𝐴𝐹))
95, 8mpbid 224 1 ((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wcel 2050  Vcvv 3415   class class class wbr 4929  dom cdm 5407  Fun wfun 6182   Fn wfn 6183  cen 8303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-en 8307
This theorem is referenced by:  tskcard  10001  hashfn  13549
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