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Theorem ffs2 30475
 Description: Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7838. (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 frnsuppeq 7838 . . 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 5914 . 2 (𝐹𝐶) = (𝐹 “ (𝐵 ∖ {𝑍}))
52, 4syl6eqr 2877 1 ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ∖ cdif 3916  {csn 4550  ◡ccnv 5541   “ cima 5545  ⟶wf 6339  (class class class)co 7149   supp csupp 7826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-supp 7827 This theorem is referenced by:  resf1o  30477  fsumcvg4  31250  eulerpartlems  31675  eulerpartlemgf  31694
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