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 | fovrnda 7434 | . . 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 4157 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | 2, 3, 4, 5, 5, 6 | off 7542 | . 2 ⊢ (𝜑 → (𝐹 ∘f + 𝐺):𝐴⟶𝐶) |
8 | 7 | frnd 6604 | 1 ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2109 ⊆ wss 3891 × cxp 5586 ran crn 5589 ⟶wf 6426 (class class class)co 7268 ∘f cof 7522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 |
This theorem is referenced by: (None) |
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