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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idreseqidinxp | Structured version Visualization version GIF version |
Description: Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.) |
Ref | Expression |
---|---|
idreseqidinxp | ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inxpssres 5687 | . . 3 ⊢ ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)) |
3 | idresssidinxp 37046 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) | |
4 | 2, 3 | eqssd 3996 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∩ cin 3944 ⊆ wss 3945 I cid 5567 × cxp 5668 ↾ cres 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-res 5682 |
This theorem is referenced by: symrefref2 37302 |
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