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Theorem disjpreima 23979
 Description: A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Assertion
Ref Expression
disjpreima Disj Disj
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem disjpreima
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inpreima 5816 . . . . . . . . 9
2 imaeq2 5158 . . . . . . . . . 10
3 ima0 5180 . . . . . . . . . 10
42, 3syl6eq 2452 . . . . . . . . 9
51, 4sylan9req 2457 . . . . . . . 8
65ex 424 . . . . . . 7
7 vex 2919 . . . . . . . . . . 11
8 csbima12g 5172 . . . . . . . . . . 11
97, 8ax-mp 8 . . . . . . . . . 10
10 csbconstg 3225 . . . . . . . . . . . 12
117, 10ax-mp 8 . . . . . . . . . . 11
1211imaeq1i 5159 . . . . . . . . . 10
139, 12eqtri 2424 . . . . . . . . 9
14 vex 2919 . . . . . . . . . . 11
15 csbima12g 5172 . . . . . . . . . . 11
1614, 15ax-mp 8 . . . . . . . . . 10
17 csbconstg 3225 . . . . . . . . . . . 12
1814, 17ax-mp 8 . . . . . . . . . . 11
1918imaeq1i 5159 . . . . . . . . . 10
2016, 19eqtri 2424 . . . . . . . . 9
2113, 20ineq12i 3500 . . . . . . . 8
2221eqeq1i 2411 . . . . . . 7
236, 22syl6ibr 219 . . . . . 6
2423orim2d 814 . . . . 5
2524ralimdv 2745 . . . 4
2625ralimdv 2745 . . 3
27 disjors 4158 . . 3 Disj
28 disjors 4158 . . 3 Disj
2926, 27, 283imtr4g 262 . 2 Disj Disj
3029imp 419 1 Disj Disj
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   wceq 1649   wcel 1721  wral 2666  cvv 2916  csb 3211   cin 3279  c0 3588  Disj wdisj 4142  ccnv 4836  cima 4840   wfun 5407 This theorem is referenced by:  sibfof  24607  dstrvprob  24682 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-disj 4143  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415
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