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Theorem ffs2 32422
Description: Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8154. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
Hypothesis
Ref Expression
ffs2.1 𝐶 = (𝐵 ∖ {𝑍})
Assertion
Ref Expression
ffs2 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))

Proof of Theorem ffs2
StepHypRef Expression
1 fsuppeq 8154 . . 3 ((𝐴𝑉𝑍𝑊) → (𝐹:𝐴𝐵 → (𝐹 supp 𝑍) = (𝐹 “ (𝐵 ∖ {𝑍}))))
213impia 1114 . 2 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹 “ (𝐵 ∖ {𝑍})))
3 ffs2.1 . . 3 𝐶 = (𝐵 ∖ {𝑍})
43imaeq2i 6047 . 2 (𝐹𝐶) = (𝐹 “ (𝐵 ∖ {𝑍}))
52, 4eqtr4di 2782 1 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cdif 3937  {csn 4620  ccnv 5665  cima 5669  wf 6529  (class class class)co 7401   supp csupp 8140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-supp 8141
This theorem is referenced by:  resf1o  32424  fsumcvg4  33419  eulerpartlems  33848  eulerpartlemgf  33867
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