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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 4634 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | reseq2i 5997 | . . 3 ⊢ ( I ↾ {𝐴, 𝐵}) = ( I ↾ ({𝐴} ∪ {𝐵})) |
3 | resundi 6014 | . . 3 ⊢ ( I ↾ ({𝐴} ∪ {𝐵})) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) | |
4 | 2, 3 | eqtri 2763 | . 2 ⊢ ( I ↾ {𝐴, 𝐵}) = (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) |
5 | xpsng 7159 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) | |
6 | 5 | anidms 566 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
8 | xpsng 7159 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) | |
9 | 8 | anidms 566 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐵} × {𝐵}) = {〈𝐵, 𝐵〉}) |
11 | 7, 10 | uneq12d 4179 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) = ({〈𝐴, 𝐴〉} ∪ {〈𝐵, 𝐵〉})) |
12 | restidsing 6073 | . . . 4 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | |
13 | restidsing 6073 | . . . 4 ⊢ ( I ↾ {𝐵}) = ({𝐵} × {𝐵}) | |
14 | 12, 13 | uneq12i 4176 | . . 3 ⊢ (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = (({𝐴} × {𝐴}) ∪ ({𝐵} × {𝐵})) |
15 | df-pr 4634 | . . 3 ⊢ {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉} = ({〈𝐴, 𝐴〉} ∪ {〈𝐵, 𝐵〉}) | |
16 | 11, 14, 15 | 3eqtr4g 2800 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (( I ↾ {𝐴}) ∪ ( I ↾ {𝐵})) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) |
17 | 4, 16 | eqtrid 2787 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( I ↾ {𝐴, 𝐵}) = {〈𝐴, 𝐴〉, 〈𝐵, 𝐵〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 {cpr 4633 〈cop 4637 I cid 5582 × cxp 5687 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: psgnprfval1 19555 |
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