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Mirrors > Home > MPE Home > Th. List > idrefALT | Structured version Visualization version GIF version |
Description: Alternate proof of idref 6908 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 3955 | . 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅)) | |
2 | elrid 5913 | . . . . . 6 ⊢ (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉) | |
3 | 2 | imbi1i 352 | . . . . 5 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅)) |
4 | r19.23v 3279 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅)) | |
5 | eleq1 2900 | . . . . . . . 8 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑦 ∈ 𝑅 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅)) | |
6 | df-br 5067 | . . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ 𝑅) | |
7 | 5, 6 | syl6bbr 291 | . . . . . . 7 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥)) |
8 | 7 | pm5.74i 273 | . . . . . 6 ⊢ ((𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
9 | 8 | ralbii 3165 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) |
10 | 3, 4, 9 | 3bitr2i 301 | . . . 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 3235 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥)) | |
13 | opex 5356 | . . . . 5 ⊢ 〈𝑥, 𝑥〉 ∈ V | |
14 | biidd 264 | . . . . 5 ⊢ (𝑦 = 〈𝑥, 𝑥〉 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥)) | |
15 | 13, 14 | ceqsalv 3532 | . . . 4 ⊢ (∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥) |
16 | 15 | ralbii 3165 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 〈𝑥, 𝑥〉 → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
17 | 11, 12, 16 | 3bitr2i 301 | . 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
18 | 1, 17 | bitri 277 | 1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 〈cop 4573 class class class wbr 5066 I cid 5459 ↾ cres 5557 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: idinxpssinxp2 35590 idinxpssinxp3 35591 symrefref3 35815 refsymrels3 35817 elrefsymrels3 35821 dfeqvrels3 35839 refrelsredund3 35884 refrelredund3 35887 |
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