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| Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2mpt | Structured version Visualization version GIF version | ||
| Description: Version of setrec2 49214 where 𝐹 is defined using maps-to notation. Deduction form is omitted in the second hypothesis for simplicity. In practice, nothing important is lost since we are only interested in one choice of 𝐴, 𝑆, and 𝑉 at a time. However, we are interested in what happens when 𝐶 varies, so deduction form is used in the third hypothesis. (Contributed by Emmett Weisz, 4-Jun-2024.) |
| Ref | Expression |
|---|---|
| setrec2mpt.1 | ⊢ 𝐵 = setrecs((𝑎 ∈ 𝐴 ↦ 𝑆)) |
| setrec2mpt.2 | ⊢ (𝑎 ∈ 𝐴 → 𝑆 ∈ 𝑉) |
| setrec2mpt.3 | ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶)) |
| Ref | Expression |
|---|---|
| setrec2mpt | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfmpt1 5250 | . 2 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ 𝑆) | |
| 2 | setrec2mpt.1 | . 2 ⊢ 𝐵 = setrecs((𝑎 ∈ 𝐴 ↦ 𝑆)) | |
| 3 | setrec2mpt.3 | . . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶)) | |
| 4 | setrec2mpt.2 | . . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → 𝑆 ∈ 𝑉) | |
| 5 | eqid 2737 | . . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 ↦ 𝑆) = (𝑎 ∈ 𝐴 ↦ 𝑆) | |
| 6 | 5 | fvmpt2 7027 | . . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆) |
| 7 | eqimss 4042 | . . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆) | |
| 8 | 6, 7 | syl 17 | . . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆) |
| 9 | 4, 8 | mpdan 687 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆) |
| 10 | 5 | fvmptndm 7047 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = ∅) |
| 11 | 0ss 4400 | . . . . . . 7 ⊢ ∅ ⊆ 𝑆 | |
| 12 | 10, 11 | eqsstrdi 4028 | . . . . . 6 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆) |
| 13 | 9, 12 | pm2.61i 182 | . . . . 5 ⊢ ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆 |
| 14 | sstr2 3990 | . . . . 5 ⊢ (((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆 → (𝑆 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶)) | |
| 15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (𝑆 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶) |
| 16 | 15 | imim2i 16 | . . 3 ⊢ ((𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶) → (𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶)) |
| 17 | 3, 16 | sylg 1823 | . 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶)) |
| 18 | 1, 2, 17 | setrec2 49214 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 setrecscsetrecs 49202 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-ne 2941 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-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-iota 6514 df-fun 6563 df-fv 6569 df-setrecs 49203 |
| This theorem is referenced by: (None) |
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