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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofrn | Structured version Visualization version GIF version |
Description: The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.) |
Ref | Expression |
---|---|
ofrn.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
ofrn.2 | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
ofrn.3 | ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶) |
ofrn.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
ofrn | ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofrn.3 | . . . 4 ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶) | |
2 | 1 | fovcdmda 7603 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐶) |
3 | ofrn.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
4 | ofrn.2 | . . 3 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
5 | ofrn.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | inidm 4234 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | 2, 3, 4, 5, 5, 6 | off 7714 | . 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝐴⟶𝐶) |
8 | 7 | frnd 6744 | 1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3962 × cxp 5686 ran crn 5689 ⟶wf 6558 (class class class)co 7430 ∘f cof 7694 |
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 |
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-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-of 7696 |
This theorem is referenced by: (None) |
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