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Theorem funimass4f 23248
 Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
Hypotheses
Ref Expression
funimass4f.1
funimass4f.2
funimass4f.3
Assertion
Ref Expression
funimass4f

Proof of Theorem funimass4f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 funimass4f.3 . . . . . . 7
21nffun 5359 . . . . . 6
3 funimass4f.1 . . . . . . 7
41nfdm 5002 . . . . . . 7
53, 4nfss 3249 . . . . . 6
62, 5nfan 1829 . . . . 5
71, 3nfima 5102 . . . . . 6
8 funimass4f.2 . . . . . 6
97, 8nfss 3249 . . . . 5
106, 9nfan 1829 . . . 4
11 funfvima2 5840 . . . . 5
12 ssel 3250 . . . . 5
1311, 12sylan9 638 . . . 4
1410, 13ralrimi 2700 . . 3
1514ex 423 . 2
163, 1dfimafnf 23247 . . . . 5
1716adantr 451 . . . 4
188abrexss 23189 . . . . 5
1918adantl 452 . . . 4
2017, 19eqsstrd 3288 . . 3
2120ex 423 . 2
2215, 21impbid 183 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1642   wcel 1710  cab 2344  wnfc 2481  wral 2619  wrex 2620   wss 3228   cdm 4771  cima 4774   wfun 5331  cfv 5337 This theorem is referenced by:  ballotlem7  24042 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
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