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Mirrors > Home > MPE Home > Th. List > fimadmfo | Structured version Visualization version GIF version |
Description: A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.) |
Ref | Expression |
---|---|
fimadmfo | β’ (πΉ:π΄βΆπ΅ β πΉ:π΄βontoβ(πΉ β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdm 6736 | . 2 β’ (πΉ:π΄βΆπ΅ β dom πΉ = π΄) | |
2 | ffn 6727 | . . . . 5 β’ (πΉ:π΄βΆπ΅ β πΉ Fn π΄) | |
3 | 2 | adantr 479 | . . . 4 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β πΉ Fn π΄) |
4 | dffn4 6822 | . . . 4 β’ (πΉ Fn π΄ β πΉ:π΄βontoβran πΉ) | |
5 | 3, 4 | sylib 217 | . . 3 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β πΉ:π΄βontoβran πΉ) |
6 | imaeq2 6064 | . . . . . . 7 β’ (π΄ = dom πΉ β (πΉ β π΄) = (πΉ β dom πΉ)) | |
7 | imadmrn 6078 | . . . . . . 7 β’ (πΉ β dom πΉ) = ran πΉ | |
8 | 6, 7 | eqtrdi 2784 | . . . . . 6 β’ (π΄ = dom πΉ β (πΉ β π΄) = ran πΉ) |
9 | 8 | eqcoms 2736 | . . . . 5 β’ (dom πΉ = π΄ β (πΉ β π΄) = ran πΉ) |
10 | 9 | adantl 480 | . . . 4 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β (πΉ β π΄) = ran πΉ) |
11 | foeq3 6814 | . . . 4 β’ ((πΉ β π΄) = ran πΉ β (πΉ:π΄βontoβ(πΉ β π΄) β πΉ:π΄βontoβran πΉ)) | |
12 | 10, 11 | syl 17 | . . 3 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β (πΉ:π΄βontoβ(πΉ β π΄) β πΉ:π΄βontoβran πΉ)) |
13 | 5, 12 | mpbird 256 | . 2 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β πΉ:π΄βontoβ(πΉ β π΄)) |
14 | 1, 13 | mpdan 685 | 1 β’ (πΉ:π΄βΆπ΅ β πΉ:π΄βontoβ(πΉ β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 dom cdm 5682 ran crn 5683 β cima 5685 Fn wfn 6548 βΆwf 6549 βontoβwfo 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fn 6556 df-f 6557 df-fo 6559 |
This theorem is referenced by: wrdsymb 14534 imasmhm 33098 imasghm 33099 imasrhm 33100 imaslmhm 33101 r1pquslmic 33322 fundcmpsurinjimaid 46798 |
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