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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 5167 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | 2 | nfrn 5827 | . 2 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐵 ↦ 𝐶) |
4 | eqid 2824 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
5 | eqid 2824 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
6 | rnmptss2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
7 | 6 | sselda 3970 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
8 | rnmptss2.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
9 | 5, 7, 8 | elrnmpt1d 41506 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
10 | 1, 3, 4, 9 | rnmptssdf 41532 | 1 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1783 ∈ wcel 2113 ⊆ wss 3939 ↦ cmpt 5149 ran crn 5559 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 |
This theorem is referenced by: smflimsuplem4 43104 |
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