Theorem ffs2 30965

Description: Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7962. (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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶))

Proof of Theorem ffs2

Step	Hyp	Ref	Expression
1		frnsuppeq 7962	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐴⟶𝐵 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍}))))
2	1	3impia 1115	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍})))
3		ffs2.1	. . 3 ⊢ 𝐶 = (𝐵 ∖ {𝑍})
4	3	imaeq2i 5956	. 2 ⊢ (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝐵 ∖ {𝑍}))
5	2, 4	eqtr4di 2797	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880  {csn 4558  ^◡ccnv 5579   “ cima 5583  ⟶wf 6414  (class class class)co 7255   supp csupp 7948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-supp 7949

This theorem is referenced by:  resf1o  30967  fsumcvg4  31802  eulerpartlems  32227  eulerpartlemgf  32246