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 6978 and djufun 6989) 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 6970). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djuinr | ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 6941 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | dff1o5 5376 | . . . . 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 6942 | . . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
6 | dff1o5 5376 | . . . . 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 3275 | . 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵)) |
10 | 1n0 6329 | . . . . 5 ⊢ 1o ≠ ∅ | |
11 | 10 | necomi 2393 | . . . 4 ⊢ ∅ ≠ 1o |
12 | disjsn2 3586 | . . . 4 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ({∅} ∩ {1o}) = ∅ |
14 | xpdisj1 4963 | . . 3 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ |
16 | 9, 15 | eqtri 2160 | 1 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ≠ wne 2308 ∩ cin 3070 ∅c0 3363 {csn 3527 × cxp 4537 ran crn 4540 ↾ cres 4541 –1-1→wf1 5120 –1-1-onto→wf1o 5122 1oc1o 6306 inlcinl 6930 inrcinr 6931 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-inl 6932 df-inr 6933 |
This theorem is referenced by: djuin 6949 casefun 6970 djudom 6978 djufun 6989 |
Copyright terms: Public domain | W3C validator |