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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptssbi | 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, 2-Jan-2022.) |
| Ref | Expression |
|---|---|
| rnmptssbi.1 | ⊢ Ⅎ𝑥𝜑 |
| rnmptssbi.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| rnmptssbi.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
| Ref | Expression |
|---|---|
| rnmptssbi | ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnmptssbi.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 2 | rnmptssbi.2 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | nfmpt1 5250 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | 2, 3 | nfcxfr 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
| 5 | 4 | nfrn 5963 | . . . . 5 ⊢ Ⅎ𝑥ran 𝐹 |
| 6 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
| 7 | 5, 6 | nfss 3976 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐶 |
| 8 | 1, 7 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ran 𝐹 ⊆ 𝐶) |
| 9 | simplr 769 | . . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → ran 𝐹 ⊆ 𝐶) | |
| 10 | simpr 484 | . . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
| 11 | rnmptssbi.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
| 12 | 11 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
| 13 | 2, 10, 12 | elrnmpt1d 5975 | . . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran 𝐹) |
| 14 | 9, 13 | sseldd 3984 | . . 3 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ 𝐶) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
| 15 | 8, 14 | ralrimia 3258 | . 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
| 16 | 2 | rnmptss 7143 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) |
| 17 | 16 | adantl 481 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ran 𝐹 ⊆ 𝐶) |
| 18 | 15, 17 | impbida 801 | 1 ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ⊆ 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: imassmpt 45269 |
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