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Theorem dfimafnf 24035
 Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
Hypotheses
Ref Expression
dfimafnf.1
dfimafnf.2
Assertion
Ref Expression
dfimafnf
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem dfimafnf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3334 . . . . . . 7
2 eqcom 2437 . . . . . . . . 9
3 funbrfvb 5761 . . . . . . . . 9
42, 3syl5bbr 251 . . . . . . . 8
54ex 424 . . . . . . 7
61, 5syl9r 69 . . . . . 6
76imp31 422 . . . . 5
87rexbidva 2714 . . . 4
98abbidv 2549 . . 3
10 dfima2 5197 . . 3
119, 10syl6reqr 2486 . 2
12 nfcv 2571 . . . 4
13 dfimafnf.1 . . . 4
14 dfimafnf.2 . . . . . 6
15 nfcv 2571 . . . . . 6
1614, 15nffv 5727 . . . . 5
1716nfeq2 2582 . . . 4
18 nfv 1629 . . . 4
19 fveq2 5720 . . . . 5
2019eqeq2d 2446 . . . 4
2112, 13, 17, 18, 20cbvrexf 2919 . . 3
2221abbii 2547 . 2
2311, 22syl6eq 2483 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2421  wnfc 2558  wrex 2698   wss 3312   class class class wbr 4204   cdm 4870  cima 4873   wfun 5440  cfv 5446 This theorem is referenced by:  funimass4f  24036 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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