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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2mpt | Structured version Visualization version GIF version |
Description: Version of setrec2 47452 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 2732 | . . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 ↦ 𝑆) = (𝑎 ∈ 𝐴 ↦ 𝑆) | |
6 | 5 | fvmpt2 6996 | . . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆) |
7 | eqimss 4037 | . . . . . . . 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 685 | . . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆) |
10 | 5 | fvmptndm 7015 | . . . . . . 7 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = ∅) |
11 | 0ss 4393 | . . . . . . 7 ⊢ ∅ ⊆ 𝑆 | |
12 | 10, 11 | eqsstrdi 4033 | . . . . . 6 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆) |
13 | 9, 12 | pm2.61i 182 | . . . . 5 ⊢ ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆 |
14 | sstr2 3986 | . . . . 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 1825 | . 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶)) |
18 | 1, 2, 17 | setrec2 47452 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ⊆ wss 3945 ∅c0 4319 ↦ cmpt 5225 ‘cfv 6533 setrecscsetrecs 47440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fv 6541 df-setrecs 47441 |
This theorem is referenced by: (None) |
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