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Theorem dom3d 7149
 Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2d.1
dom2d.2
dom3d.3
dom3d.4
Assertion
Ref Expression
dom3d
Distinct variable groups:   ,,   ,,   ,   ,   ,,
Allowed substitution hints:   ()   ()   (,)   (,)

Proof of Theorem dom3d
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dom2d.1 . . . . . 6
2 dom2d.2 . . . . . 6
31, 2dom2lem 7147 . . . . 5
4 f1f 5639 . . . . 5
53, 4syl 16 . . . 4
6 dom3d.3 . . . 4
7 dom3d.4 . . . 4
8 fex2 5603 . . . 4
95, 6, 7, 8syl3anc 1184 . . 3
10 f1eq1 5634 . . . 4
1110spcegv 3037 . . 3
129, 3, 11sylc 58 . 2
13 brdomg 7118 . . 3
147, 13syl 16 . 2
1512, 14mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  cvv 2956   class class class wbr 4212   cmpt 4266  wf 5450  wf1 5451   cdom 7107 This theorem is referenced by:  dom3  7151  xpdom2  7203  fopwdom  7216 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fv 5462  df-dom 7111
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