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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 7087 | . 2 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) | |
2 | djurclr 7080 | . 2 ⊢ (𝑥 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑥) ∈ (𝐴 ⊔ 𝐵)) | |
3 | 1, 2 | inresflem 7090 | 1 ⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↾ cres 4646 –1-1→wf1 5232 1oc1o 6435 ⊔ cdju 7067 inrcinr 7076 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-1st 6166 df-2nd 6167 df-1o 6442 df-dju 7068 df-inr 7078 |
This theorem is referenced by: updjudhcoinrg 7111 updjud 7112 caserel 7117 djudom 7123 djufun 7134 djuinj 7136 djudomr 7250 exmidsbthrlem 15249 |
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