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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptssd | Structured version Visualization version GIF version | ||
| Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
| Ref | Expression |
|---|---|
| rnmptssd.1 | ⊢ Ⅎ𝑥𝜑 |
| rnmptssd.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| rnmptssd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
| Ref | Expression |
|---|---|
| rnmptssd | ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnmptssd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | rnmptssd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
| 3 | 1, 2 | ralrimia 3258 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
| 4 | rnmptssd.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 5 | 4 | rnmptss 7143 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
| 6 | 3, 5 | syl 17 | 1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: infnsuprnmpt 45257 suprclrnmpt 45258 suprubrnmpt2 45259 suprubrnmpt 45260 fisupclrnmpt 45409 supxrleubrnmpt 45417 infxrlbrnmpt2 45421 supxrrernmpt 45432 suprleubrnmpt 45433 infrnmptle 45434 infxrunb3rnmpt 45439 supxrre3rnmpt 45440 supminfrnmpt 45456 infxrrnmptcl 45458 infxrgelbrnmpt 45465 infrpgernmpt 45476 supminfxrrnmpt 45482 liminfcl 45778 fourierdlem31 46153 fourierdlem53 46174 sge0xaddlem2 46449 sge0reuz 46462 sge0reuzb 46463 meadjiun 46481 hoidmvlelem2 46611 iunhoiioolem 46690 vonioolem1 46695 smflimsuplem4 46838 |
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