Theorem ffs2 32815

Description: Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8118. (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		fsuppeq 8118	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐴⟶𝐵 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍}))))
2	1	3impia 1118	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍})))
3		ffs2.1	. . 3 ⊢ 𝐶 = (𝐵 ∖ {𝑍})
4	3	imaeq2i 6017	. 2 ⊢ (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝐵 ∖ {𝑍}))
5	2, 4	eqtr4di 2790	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887  {csn 4568  ^◡ccnv 5623   “ cima 5627  ⟶wf 6488  (class class class)co 7360   supp csupp 8103

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-supp 8104

This theorem is referenced by:  resf1o  32818  fsumcvg4  34110  eulerpartlems  34520  eulerpartlemgf  34539