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Theorem residpr 6315
 Description: Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)
Assertion
Ref Expression
residpr ((𝐴𝑉𝐵𝑊) → ( I ↾ {𝐴, 𝐵}) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})

Proof of Theorem residpr
StepHypRef Expression
1 df-pr 4128 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
21reseq2i 5314 . . 3 ( I ↾ {𝐴, 𝐵}) = ( I ↾ ({𝐴} ∪ {𝐵}))
3 resundi 5330 . . 3 ( I ↾ ({𝐴} ∪ {𝐵})) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵}))
42, 3eqtri 2632 . 2 ( I ↾ {𝐴, 𝐵}) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵}))
5 xpsng 6312 . . . . . 6 ((𝐴𝑉𝐴𝑉) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
65anidms 675 . . . . 5 (𝐴𝑉 → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
76adantr 480 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩})
8 xpsng 6312 . . . . . 6 ((𝐵𝑊𝐵𝑊) → ({𝐵} × {𝐵}) = {⟨𝐵, 𝐵⟩})
98anidms 675 . . . . 5 (𝐵𝑊 → ({𝐵} × {𝐵}) = {⟨𝐵, 𝐵⟩})
109adantl 481 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐵} × {𝐵}) = {⟨𝐵, 𝐵⟩})
117, 10uneq12d 3730 . . 3 ((𝐴𝑉𝐵𝑊) → (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) = ({⟨𝐴, 𝐴⟩} ∪ {⟨𝐵, 𝐵⟩}))
12 restidsing 5377 . . . 4 ( I ↾ {𝐴}) = ({𝐴} × {𝐴})
13 restidsing 5377 . . . 4 ( I ↾ {𝐵}) = ({𝐵} × {𝐵})
1412, 13uneq12i 3727 . . 3 (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵}))
15 df-pr 4128 . . 3 {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩} = ({⟨𝐴, 𝐴⟩} ∪ {⟨𝐵, 𝐵⟩})
1611, 14, 153eqtr4g 2669 . 2 ((𝐴𝑉𝐵𝑊) → (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})
174, 16syl5eq 2656 1 ((𝐴𝑉𝐵𝑊) → ( I ↾ {𝐴, 𝐵}) = {⟨𝐴, 𝐴⟩, ⟨𝐵, 𝐵⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {csn 4125  {cpr 4127  ⟨cop 4131   I cid 4948   × cxp 5036   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  psgnprfval1  17765
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