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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 6678 | . 2 β’ (πΉ:π΄βΆπ΅ β dom πΉ = π΄) | |
2 | ffn 6669 | . . . . 5 β’ (πΉ:π΄βΆπ΅ β πΉ Fn π΄) | |
3 | 2 | adantr 482 | . . . 4 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β πΉ Fn π΄) |
4 | dffn4 6763 | . . . 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 6010 | . . . . . . 7 β’ (π΄ = dom πΉ β (πΉ β π΄) = (πΉ β dom πΉ)) | |
7 | imadmrn 6024 | . . . . . . 7 β’ (πΉ β dom πΉ) = ran πΉ | |
8 | 6, 7 | eqtrdi 2793 | . . . . . 6 β’ (π΄ = dom πΉ β (πΉ β π΄) = ran πΉ) |
9 | 8 | eqcoms 2745 | . . . . 5 β’ (dom πΉ = π΄ β (πΉ β π΄) = ran πΉ) |
10 | 9 | adantl 483 | . . . 4 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β (πΉ β π΄) = ran πΉ) |
11 | foeq3 6755 | . . . 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 257 | . 2 β’ ((πΉ:π΄βΆπ΅ β§ dom πΉ = π΄) β πΉ:π΄βontoβ(πΉ β π΄)) |
14 | 1, 13 | mpdan 686 | 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 397 = wceq 1542 dom cdm 5634 ran crn 5635 β cima 5637 Fn wfn 6492 βΆwf 6493 βontoβwfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fn 6500 df-f 6501 df-fo 6503 |
This theorem is referenced by: wrdsymb 14431 fundcmpsurinjimaid 45610 |
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