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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 | fovrnda 7040 | . . 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 4019 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
7 | 2, 3, 4, 5, 5, 6 | off 7147 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺):𝐴⟶𝐶) |
8 | 7 | frnd 6264 | 1 ⊢ (𝜑 → ran (𝐹 ∘𝑓 + 𝐺) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 ⊆ wss 3770 × cxp 5311 ran crn 5314 ⟶wf 6098 (class class class)co 6879 ∘𝑓 cof 7130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 |
This theorem is referenced by: (None) |
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