Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptssdf | 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 |
---|---|
rnmptssdf.1 | ⊢ Ⅎ𝑥𝜑 |
rnmptssdf.2 | ⊢ Ⅎ𝑥𝐶 |
rnmptssdf.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
rnmptssdf.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
rnmptssdf | ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmptssdf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rnmptssdf.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
3 | 1, 2 | ralrimia 3428 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
4 | rnmptssdf.2 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
5 | rnmptssdf.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 4, 5 | rnmptssf 42746 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
7 | 3, 6 | syl 17 | 1 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 Ⅎwnf 1789 ∈ wcel 2109 Ⅎwnfc 2888 ∀wral 3065 ⊆ wss 3891 ↦ cmpt 5161 ran crn 5589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-fun 6432 df-fn 6433 df-f 6434 |
This theorem is referenced by: rnmptss2 42756 supminfrnmpt 42939 supminfxrrnmpt 42965 |
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