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| Mirrors > Home > MPE Home > Th. List > mptimass | Structured version Visualization version GIF version | ||
| Description: Image of a function in maps-to notation for a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| Ref | Expression |
|---|---|
| mptimass.1 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
| Ref | Expression |
|---|---|
| mptimass | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mptima 6090 | . 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) | |
| 2 | mptimass.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
| 3 | sseqin2 4223 | . . . . 5 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶) | |
| 4 | 2, 3 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐶) = 𝐶) |
| 5 | 4 | mpteq1d 5237 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)) |
| 6 | 5 | rneqd 5949 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ 𝐵) = ran (𝑥 ∈ 𝐶 ↦ 𝐵)) |
| 7 | 1, 6 | eqtrid 2789 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 “ cima 5688 |
| 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-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-clab 2715 df-cleq 2729 df-clel 2816 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-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: pzriprnglem10 21501 limsupresico 45715 limsupvaluz 45723 liminfresico 45786 |
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