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Mirrors > Home > ILE Home > Th. List > inlresf1 | GIF version |
Description: The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
inlresf1 | ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 7031 | . 2 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) | |
2 | djulclr 7024 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
3 | 1, 2 | inresflem 7035 | 1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∅c0 3414 ↾ cres 4611 –1-1→wf1 5193 ⊔ cdju 7012 inlcinl 7020 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-1st 6117 df-2nd 6118 df-dju 7013 df-inl 7022 |
This theorem is referenced by: updjudhcoinlf 7055 updjud 7057 caserel 7062 djudom 7068 difinfsn 7075 djufun 7079 djuinj 7081 djudoml 7189 |
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