Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > residpr | Structured version Visualization version GIF version |
Description: Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.) |
Ref | Expression |
---|---|
residpr | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( I ↾ {𝐴, 𝐵}) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4128 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | reseq2i 5314 | . . 3 ⊢ ( I ↾ {𝐴, 𝐵}) = ( I ↾ ({𝐴} ∪ {𝐵})) |
3 | resundi 5330 | . . 3 ⊢ ( I ↾ ({𝐴} ∪ {𝐵})) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) | |
4 | 2, 3 | eqtri 2632 | . 2 ⊢ ( I ↾ {𝐴, 𝐵}) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) |
5 | xpsng 6312 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) | |
6 | 5 | anidms 675 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
8 | xpsng 6312 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) | |
9 | 8 | anidms 675 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) |
11 | 7, 10 | uneq12d 3730 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) = ({〈𝐴, 𝐴〉} ∪ {〈𝐵, 𝐵〉})) |
12 | restidsing 5377 | . . . 4 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | |
13 | restidsing 5377 | . . . 4 ⊢ ( I ↾ {𝐵}) = ({𝐵} × {𝐵}) | |
14 | 12, 13 | uneq12i 3727 | . . 3 ⊢ (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) |
15 | df-pr 4128 | . . 3 ⊢ {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉} = ({〈𝐴, 𝐴〉} ∪ {〈𝐵, 𝐵〉}) | |
16 | 11, 14, 15 | 3eqtr4g 2669 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) |
17 | 4, 16 | syl5eq 2656 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( 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 |
Copyright terms: Public domain | W3C validator |