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Mathbox for Thierry Arnoux |
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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 codomain rather than the universal class. See also fsuppeq 8159. (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 | fsuppeq 8159 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐴⟶𝐵 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍})))) | |
2 | 1 | 3impia 1117 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝐵 ∖ {𝑍}))) |
3 | ffs2.1 | . . 3 ⊢ 𝐶 = (𝐵 ∖ {𝑍}) | |
4 | 3 | imaeq2i 6057 | . 2 ⊢ (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝐵 ∖ {𝑍})) |
5 | 2, 4 | eqtr4di 2790 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∖ cdif 3945 {csn 4628 ◡ccnv 5675 “ cima 5679 ⟶wf 6539 (class class class)co 7408 supp csupp 8145 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-supp 8146 |
This theorem is referenced by: resf1o 31950 fsumcvg4 32925 eulerpartlems 33354 eulerpartlemgf 33373 |
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