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Mirrors > Home > ILE Home > Th. List > f1oi | GIF version |
Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1oi | ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 5300 | . 2 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
2 | cnvresid 5257 | . . . 4 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
3 | 2 | fneq1i 5277 | . . 3 ⊢ (◡( I ↾ 𝐴) Fn 𝐴 ↔ ( I ↾ 𝐴) Fn 𝐴) |
4 | 1, 3 | mpbir 145 | . 2 ⊢ ◡( I ↾ 𝐴) Fn 𝐴 |
5 | dff1o4 5435 | . 2 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴) Fn 𝐴 ∧ ◡( I ↾ 𝐴) Fn 𝐴)) | |
6 | 1, 4, 5 | mpbir2an 931 | 1 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 |
Colors of variables: wff set class |
Syntax hints: I cid 4261 ◡ccnv 4598 ↾ cres 4601 Fn wfn 5178 –1-1-onto→wf1o 5182 |
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-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-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 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-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 |
This theorem is referenced by: f1ovi 5466 isoid 5773 enrefg 6722 ssdomg 6736 omp1eomlem 7051 ctm 7066 omct 7074 ctssexmid 7106 ssomct 12341 ssidcn 12777 dvid 13229 dvexp 13242 subctctexmid 13743 |
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