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Mirrors > Home > ILE Home > Th. List > inrresf1 | GIF version |
Description: The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
inrresf1 | ⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djurf1or 6747 | . 2 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1𝑜} × 𝐵) | |
2 | djurclr 6740 | . 2 ⊢ (𝑥 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
3 | 1, 2 | inresflem 6750 | 1 ⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↾ cres 4440 –1-1→wf1 5012 1𝑜c1o 6174 ⊔ cdju 6728 inrcinr 6736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-1st 5911 df-2nd 5912 df-1o 6181 df-dju 6729 df-inr 6738 |
This theorem is referenced by: djuun 6758 updjudhcoinrg 6770 updjud 6771 caserel 6776 djufun 6782 djuinj 6784 djudom 6785 exmidsbthrlem 11867 |
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