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 4553 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | reseq2i 5857 | . . 3 ⊢ ( I ↾ {𝐴, 𝐵}) = ( I ↾ ({𝐴} ∪ {𝐵})) |
3 | resundi 5874 | . . 3 ⊢ ( I ↾ ({𝐴} ∪ {𝐵})) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) | |
4 | 2, 3 | eqtri 2766 | . 2 ⊢ ( I ↾ {𝐴, 𝐵}) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) |
5 | xpsng 6963 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) | |
6 | 5 | anidms 570 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
7 | 6 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
8 | xpsng 6963 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) | |
9 | 8 | anidms 570 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) |
11 | 7, 10 | uneq12d 4087 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) = ({〈𝐴, 𝐴〉} ∪ {〈𝐵, 𝐵〉})) |
12 | restidsing 5931 | . . . 4 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | |
13 | restidsing 5931 | . . . 4 ⊢ ( I ↾ {𝐵}) = ({𝐵} × {𝐵}) | |
14 | 12, 13 | uneq12i 4084 | . . 3 ⊢ (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) |
15 | df-pr 4553 | . . 3 ⊢ {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉} = ({〈𝐴, 𝐴〉} ∪ {〈𝐵, 𝐵〉}) | |
16 | 11, 14, 15 | 3eqtr4g 2804 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) |
17 | 4, 16 | eqtrid 2790 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( I ↾ {𝐴, 𝐵}) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∪ cun 3873 {csn 4550 {cpr 4552 〈cop 4556 I cid 5463 × cxp 5558 ↾ cres 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 |
This theorem is referenced by: psgnprfval1 18927 |
Copyright terms: Public domain | W3C validator |