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Theorem suppeqfsuppbi 7863
Description: If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.)
Assertion
Ref Expression
suppeqfsuppbi

Proof of Theorem suppeqfsuppbi
StepHypRef Expression
1 simprlr 764 . . . . . . 7
2 simprll 763 . . . . . . 7
3 simpl 457 . . . . . . 7
4 funisfsupp 7854 . . . . . . 7
51, 2, 3, 4syl3anc 1228 . . . . . 6
65adantr 465 . . . . 5
7 simpr 461 . . . . . . . . . . 11
87adantr 465 . . . . . . . . . 10
9 simpl 457 . . . . . . . . . . 11
109adantr 465 . . . . . . . . . 10
11 simpr 461 . . . . . . . . . 10
12 funisfsupp 7854 . . . . . . . . . 10
138, 10, 11, 12syl3anc 1228 . . . . . . . . 9
1413ex 434 . . . . . . . 8
1514adantl 466 . . . . . . 7
1615impcom 430 . . . . . 6
17 eleq1 2529 . . . . . . 7
1817bicomd 201 . . . . . 6
1916, 18sylan9bb 699 . . . . 5
206, 19bitr4d 256 . . . 4
2120ex 434 . . 3
2221ex 434 . 2
23 relfsupp 7851 . . . . . 6
2423brrelex2i 5046 . . . . 5
2523brrelex2i 5046 . . . . 5
2624, 25pm5.21ni 352 . . . 4
2726a1d 25 . . 3
2827a1d 25 . 2
2922, 28pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109   class class class wbr 4452  Funwfun 5587  (class class class)co 6296   csupp 6918   cfn 7536   cfsupp 7849
This theorem is referenced by:  cantnfrescl  8116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-fsupp 7850
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