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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 7091 and djufun 7102) 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 7083). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djuinr | ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 7054 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | dff1o5 5470 | . . . . 5 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) ↔ ((inl ↾ 𝐴):𝐴–1-1→({∅} × 𝐴) ∧ ran (inl ↾ 𝐴) = ({∅} × 𝐴))) | |
3 | 2 | simprbi 275 | . . . 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 7055 | . . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
6 | dff1o5 5470 | . . . . 5 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) ↔ ((inr ↾ 𝐵):𝐵–1-1→({1o} × 𝐵) ∧ ran (inr ↾ 𝐵) = ({1o} × 𝐵))) | |
7 | 6 | simprbi 275 | . . . 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 3334 | . 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵)) |
10 | 1n0 6432 | . . . . 5 ⊢ 1o ≠ ∅ | |
11 | 10 | necomi 2432 | . . . 4 ⊢ ∅ ≠ 1o |
12 | disjsn2 3655 | . . . 4 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ({∅} ∩ {1o}) = ∅ |
14 | xpdisj1 5053 | . . 3 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ |
16 | 9, 15 | eqtri 2198 | 1 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ≠ wne 2347 ∩ cin 3128 ∅c0 3422 {csn 3592 × cxp 4624 ran crn 4627 ↾ cres 4628 –1-1→wf1 5213 –1-1-onto→wf1o 5215 1oc1o 6409 inlcinl 7043 inrcinr 7044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-1st 6140 df-2nd 6141 df-1o 6416 df-inl 7045 df-inr 7046 |
This theorem is referenced by: djuin 7062 casefun 7083 djudom 7091 djufun 7102 |
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