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Mirrors > Home > MPE Home > Th. List > Mathboxes > ffs2 | Structured version Visualization version GIF version |
Description: Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7842. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
Ref | Expression |
---|---|
ffs2.1 | ⊢ 𝐶 = (𝐵 ∖ {𝑍}) |
Ref | Expression |
---|---|
ffs2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frnsuppeq 7842 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐴⟶𝐵 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍})))) | |
2 | 1 | 3impia 1113 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍}))) |
3 | ffs2.1 | . . 3 ⊢ 𝐶 = (𝐵 ∖ {𝑍}) | |
4 | 3 | imaeq2i 5927 | . 2 ⊢ (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝐵 ∖ {𝑍})) |
5 | 2, 4 | syl6eqr 2874 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 {csn 4567 ◡ccnv 5554 “ cima 5558 ⟶wf 6351 (class class class)co 7156 supp csupp 7830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-supp 7831 |
This theorem is referenced by: resf1o 30466 fsumcvg4 31193 eulerpartlems 31618 eulerpartlemgf 31637 |
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