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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 7049 and djufun 7060) 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 7041). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djuinr | ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 7012 | . . . 4 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | dff1o5 5435 | . . . . 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 7013 | . . . 4 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
6 | dff1o5 5435 | . . . . 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 3316 | . 2 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = (({∅} × 𝐴) ∩ ({1o} × 𝐵)) |
10 | 1n0 6391 | . . . . 5 ⊢ 1o ≠ ∅ | |
11 | 10 | necomi 2419 | . . . 4 ⊢ ∅ ≠ 1o |
12 | disjsn2 3633 | . . . 4 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ({∅} ∩ {1o}) = ∅ |
14 | xpdisj1 5022 | . . 3 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ |
16 | 9, 15 | eqtri 2185 | 1 ⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ≠ wne 2334 ∩ cin 3110 ∅c0 3404 {csn 3570 × cxp 4596 ran crn 4599 ↾ cres 4600 –1-1→wf1 5179 –1-1-onto→wf1o 5181 1oc1o 6368 inlcinl 7001 inrcinr 7002 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-1o 6375 df-inl 7003 df-inr 7004 |
This theorem is referenced by: djuin 7020 casefun 7041 djudom 7049 djufun 7060 |
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