Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > djuinr | GIF version |
Description: The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 6971 and djufun 6982) while the simpler statement ⊢ (ran inl ∩ ran inr) = ∅ is easily recovered from it by substituting V for both 𝐴 and 𝐵 as done in casefun 6963). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djuinr | ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 6934 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | dff1o5 5369 | . . . . 5 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) ↔ ((inl ↾ 𝐴):𝐴–1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴))) | |
3 | 2 | simprbi 273 | . . . 4 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → ran (inl ↾ 𝐴) = ({∅} × 𝐴)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ran (inl ↾ 𝐴) = ({∅} × 𝐴) |
5 | djurf1or 6935 | . . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
6 | dff1o5 5369 | . . . . 5 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) ↔ ((inr ↾ 𝐵):𝐵–1-1→({1o} × 𝐵) ∧ ran (inr ↾ 𝐵) = ({1o} × 𝐵))) | |
7 | 6 | simprbi 273 | . . . 4 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → ran (inr ↾ 𝐵) = ({1o} × 𝐵)) |
8 | 5, 7 | ax-mp 5 | . . 3 ⊢ ran (inr ↾ 𝐵) = ({1o} × 𝐵) |
9 | 4, 8 | ineq12i 3270 | . 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵)) |
10 | 1n0 6322 | . . . . 5 ⊢ 1o ≠ ∅ | |
11 | 10 | necomi 2391 | . . . 4 ⊢ ∅ ≠ 1o |
12 | disjsn2 3581 | . . . 4 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ({∅} ∩ {1o}) = ∅ |
14 | xpdisj1 4958 | . . 3 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ |
16 | 9, 15 | eqtri 2158 | 1 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ≠ wne 2306 ∩ cin 3065 ∅c0 3358 {csn 3522 × cxp 4532 ran crn 4535 ↾ cres 4536 –1-1→wf1 5115 –1-1-onto→wf1o 5117 1oc1o 6299 inlcinl 6923 inrcinr 6924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1st 6031 df-2nd 6032 df-1o 6306 df-inl 6925 df-inr 6926 |
This theorem is referenced by: djuin 6942 casefun 6963 djudom 6971 djufun 6982 |
Copyright terms: Public domain | W3C validator |