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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptss2 | Structured version Visualization version GIF version |
Description: The range of an operation 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 5125 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | 2 | nfrn 5789 | . 2 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
4 | eqid 2738 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
5 | eqid 2738 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
6 | rnmptss2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | 6 | sselda 3875 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
8 | rnmptss2.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
9 | 5, 7, 8 | elrnmpt1d 42295 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
10 | 1, 3, 4, 9 | rnmptssdf 42321 | 1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 Ⅎwnf 1790 ∈ wcel 2113 ⊆ wss 3841 ↦ cmpt 5107 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6335 df-fn 6336 df-f 6337 |
This theorem is referenced by: smflimsuplem4 43879 |
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