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Theorem dfimafnf 23041
 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 3174 . . . . . . 7
2 eqcom 2285 . . . . . . . . 9
3 funbrfvb 5565 . . . . . . . . 9
42, 3syl5bbr 250 . . . . . . . 8
54ex 423 . . . . . . 7
61, 5syl9r 67 . . . . . 6
76imp31 421 . . . . 5
87rexbidva 2560 . . . 4
98abbidv 2397 . . 3
10 dfima2 5014 . . 3
119, 10syl6reqr 2334 . 2
12 nfcv 2419 . . . 4
13 dfimafnf.1 . . . 4
14 nfcv 2419 . . . . 5
15 dfimafnf.2 . . . . . 6
16 nfcv 2419 . . . . . 6
1715, 16nffv 5532 . . . . 5
1814, 17nfeq 2426 . . . 4
19 nfv 1605 . . . 4
20 fveq2 5525 . . . . 5
2120eqeq2d 2294 . . . 4
2212, 13, 18, 19, 21cbvrexf 2759 . . 3
2322abbii 2395 . 2
2411, 23syl6eq 2331 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  cab 2269  wnfc 2406  wrex 2544   wss 3152   class class class wbr 4023   cdm 4689  cima 4692   wfun 5249  cfv 5255 This theorem is referenced by:  funimass4f  23042 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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