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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 5694 | . . 3 ⊢ ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)) |
3 | idresssidinxp 37481 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) | |
4 | 2, 3 | eqssd 4000 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∩ cin 3948 ⊆ wss 3949 I cid 5574 × cxp 5675 ↾ cres 5679 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-res 5689 |
This theorem is referenced by: symrefref2 37737 |
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