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Mirrors > Home > MPE Home > Th. List > idrefALT | Structured version Visualization version GIF version |
Description: Alternate proof of idref 7146 not relying on definitions related to functions. Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) (Proof shortened by BJ, 28-Aug-2022.) The "proof modification is discouraged" tag is here only because this is an *ALT result. (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
idrefALT | ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3968 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) | |
2 | elrid 6045 | . . . . . 6 ⊢ (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉) | |
3 | 2 | imbi1i 349 | . . . . 5 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅)) |
4 | r19.23v 3181 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅)) | |
5 | eleq1 2820 | . . . . . . . 8 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑦 ∈ 𝑅 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅)) | |
6 | df-br 5149 | . . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
7 | 5, 6 | bitr4di 289 | . . . . . . 7 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥)) |
8 | 7 | pm5.74i 271 | . . . . . 6 ⊢ ((𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
9 | 8 | ralbii 3092 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
10 | 3, 4, 9 | 3bitr2i 299 | . . . 4 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
11 | 10 | albii 1820 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
12 | ralcom4 3282 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) | |
13 | opex 5464 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V | |
14 | biidd 262 | . . . . 5 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥)) | |
15 | 13, 14 | ceqsalv 3511 | . . . 4 ⊢ (∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥) |
16 | 15 | ralbii 3092 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
17 | 11, 12, 16 | 3bitr2i 299 | . 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
18 | 1, 17 | bitri 275 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1538 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ⊆ wss 3948 〈cop 4634 class class class wbr 5148 I cid 5573 ↾ cres 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-11 2153 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-res 5688 |
This theorem is referenced by: idinxpssinxp2 37651 idinxpssinxp3 37652 symrefref3 37898 refsymrels3 37900 elrefsymrels3 37904 dfeqvrels3 37923 refrelsredund3 37968 refrelredund3 37971 |
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