Theorem ffs2 32879

Description: Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8150. (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 8150	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐴⟶𝐵 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍}))))
2	1	3impia 1129	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍})))
3		ffs2.1	. . 3 ⊢ 𝐶 = (𝐵 ∖ {𝑍})
4	3	imaeq2i 6044	. 2 ⊢ (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝐵 ∖ {𝑍}))
5	2, 4	eqtr4di 2814	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∖ cdif 3901  {csn 4581  ^◡ccnv 5644   “ cima 5648  ⟶wf 6513  (class class class)co 7392   supp csupp 8135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-supp 8136

This theorem is referenced by:  resf1o  32882  fsumcvg4  34208  eulerpartlems  34618  eulerpartlemgf  34637