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Mirrors > Home > MPE Home > Th. List > opelidres | Structured version Visualization version GIF version |
Description: 〈𝐴, 𝐴〉 belongs to a restriction of the identity class iff 𝐴 belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.) |
Ref | Expression |
---|---|
opelidres | ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ididg 5727 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴) | |
2 | df-br 5070 | . . 3 ⊢ (𝐴 I 𝐴 ↔ 〈𝐴, 𝐴〉 ∈ I ) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 ∈ I ) |
4 | opelres 5862 | . 2 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ (𝐴 ∈ 𝐵 ∧ 〈𝐴, 𝐴〉 ∈ I ))) | |
5 | 3, 4 | mpbiran2d 706 | 1 ⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 ∈ ( I ↾ 𝐵) ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2113 〈cop 4576 class class class wbr 5069 I cid 5462 ↾ cres 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-res 5570 |
This theorem is referenced by: ustfilxp 22824 ustelimasn 22834 metustfbas 23170 dfpo2 32995 |
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