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Mirrors > Home > MPE Home > Th. List > idrefALT | Structured version Visualization version GIF version |
Description: Alternate proof of idref 6918 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 3863 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) | |
2 | elrid 5887 | . . . . . 6 ⊢ (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉) | |
3 | 2 | imbi1i 353 | . . . . 5 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅)) |
4 | r19.23v 3189 | . . . . 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 5031 | . . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
7 | 5, 6 | bitr4di 292 | . . . . . . 7 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥)) |
8 | 7 | pm5.74i 274 | . . . . . 6 ⊢ ((𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
9 | 8 | ralbii 3080 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
10 | 3, 4, 9 | 3bitr2i 302 | . . . 4 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
11 | 10 | albii 1826 | . . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
12 | ralcom4 3148 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) | |
13 | opex 5322 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V | |
14 | biidd 265 | . . . . 5 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥)) | |
15 | 13, 14 | ceqsalv 3433 | . . . 4 ⊢ (∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥) |
16 | 15 | ralbii 3080 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
17 | 11, 12, 16 | 3bitr2i 302 | . 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
18 | 1, 17 | bitri 278 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 ⊆ wss 3843 〈cop 4522 class class class wbr 5030 I cid 5428 ↾ cres 5527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-res 5537 |
This theorem is referenced by: idinxpssinxp2 36076 idinxpssinxp3 36077 symrefref3 36301 refsymrels3 36303 elrefsymrels3 36307 dfeqvrels3 36325 refrelsredund3 36370 refrelredund3 36373 |
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