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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsuppfnd | Structured version Visualization version GIF version |
Description: Deduce membership in the support of a function. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
Ref | Expression |
---|---|
elsuppfnd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
elsuppfnd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elsuppfnd.3 | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
elsuppfnd.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
elsuppfnd.5 | ⊢ (𝜑 → (𝐹‘𝑋) ≠ 𝑍) |
Ref | Expression |
---|---|
elsuppfnd | ⊢ (𝜑 → 𝑋 ∈ (𝐹 supp 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuppfnd.1 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | elsuppfnd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | elsuppfnd.3 | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
4 | elsuppfnd.4 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
5 | elsuppfnd.5 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ≠ 𝑍) | |
6 | elsuppfn 8193 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
7 | 6 | biimpar 477 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍)) → 𝑋 ∈ (𝐹 supp 𝑍)) |
8 | 1, 2, 3, 4, 5, 7 | syl32anc 1377 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐹 supp 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2105 ≠ wne 2937 Fn wfn 6557 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-supp 8184 |
This theorem is referenced by: elrgspnlem2 33232 |
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