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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptss2 | 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, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| rnmptss2.1 | ⊢ Ⅎ𝑥𝜑 |
| rnmptss2.3 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| rnmptss2.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
| Ref | Expression |
|---|---|
| rnmptss2 | ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnmptss2.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | nfmpt1 5250 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶) | |
| 3 | 2 | nfrn 5963 | . 2 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
| 4 | eqid 2737 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
| 5 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
| 6 | rnmptss2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 7 | 6 | sselda 3983 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
| 8 | rnmptss2.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
| 9 | 5, 7, 8 | elrnmpt1d 5975 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
| 10 | 1, 3, 4, 9 | rnmptssdf 45261 | 1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 ⊆ 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-sbc 3789 df-csb 3900 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: smflimsuplem4 46838 |
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