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Mirrors > Home > ILE Home > Th. List > djurf1or | GIF version |
Description: The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) |
Ref | Expression |
---|---|
djurf1or | ⊢ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 6203 | . 2 ⊢ 1o ∈ V | |
2 | df-inr 6794 | . 2 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
3 | 1, 2 | djuf1olemr 6800 | 1 ⊢ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴) |
Colors of variables: wff set class |
Syntax hints: {csn 3450 × cxp 4449 ↾ cres 4453 –1-1-onto→wf1o 5027 1oc1o 6188 inrcinr 6792 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-1st 5925 df-2nd 5926 df-1o 6195 df-inr 6794 |
This theorem is referenced by: inrresf1 6808 djuinr 6809 djuunr 6812 eldju 6813 |
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