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Mirrors > Home > MPE Home > Th. List > idrefALT | Structured version Visualization version GIF version |
Description: Alternate proof of idref 7166 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 | df-ss 3980 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) | |
2 | elrid 6066 | . . . . . 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 2827 | . . . . . . . 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 3091 | . . . . 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 1816 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
12 | ralcom4 3284 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) | |
13 | opex 5475 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V | |
14 | biidd 262 | . . . . 5 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥)) | |
15 | 13, 14 | ceqsalv 3519 | . . . 4 ⊢ (∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥) |
16 | 15 | ralbii 3091 | . . 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 206 ∀wal 1535 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 〈cop 4637 class class class wbr 5148 I cid 5582 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: idinxpssinxp2 38300 idinxpssinxp3 38301 symrefref3 38546 refsymrels3 38548 elrefsymrels3 38552 dfeqvrels3 38571 refrelsredund3 38616 refrelredund3 38619 |
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