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Mirrors > Home > MPE Home > Th. List > fco3OLD | Structured version Visualization version GIF version |
Description: Obsolete version of funcofd 6743 as 20-Sep-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fco3OLD.1 | β’ (π β Fun πΉ) |
fco3OLD.2 | β’ (π β Fun πΊ) |
Ref | Expression |
---|---|
fco3OLD | β’ (π β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fco3OLD.1 | . . . . 5 β’ (π β Fun πΉ) | |
2 | fco3OLD.2 | . . . . 5 β’ (π β Fun πΊ) | |
3 | funco 6581 | . . . . 5 β’ ((Fun πΉ β§ Fun πΊ) β Fun (πΉ β πΊ)) | |
4 | 1, 2, 3 | syl2anc 583 | . . . 4 β’ (π β Fun (πΉ β πΊ)) |
5 | fdmrn 6742 | . . . 4 β’ (Fun (πΉ β πΊ) β (πΉ β πΊ):dom (πΉ β πΊ)βΆran (πΉ β πΊ)) | |
6 | 4, 5 | sylib 217 | . . 3 β’ (π β (πΉ β πΊ):dom (πΉ β πΊ)βΆran (πΉ β πΊ)) |
7 | dmco 6246 | . . . 4 β’ dom (πΉ β πΊ) = (β‘πΊ β dom πΉ) | |
8 | 7 | feq2i 6702 | . . 3 β’ ((πΉ β πΊ):dom (πΉ β πΊ)βΆran (πΉ β πΊ) β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran (πΉ β πΊ)) |
9 | 6, 8 | sylib 217 | . 2 β’ (π β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran (πΉ β πΊ)) |
10 | rncoss 5964 | . . 3 β’ ran (πΉ β πΊ) β ran πΉ | |
11 | 10 | a1i 11 | . 2 β’ (π β ran (πΉ β πΊ) β ran πΉ) |
12 | 9, 11 | fssd 6728 | 1 β’ (π β (πΉ β πΊ):(β‘πΊ β dom πΉ)βΆran πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wss 3943 β‘ccnv 5668 dom cdm 5669 ran crn 5670 β cima 5672 β ccom 5673 Fun wfun 6530 βΆwf 6532 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6538 df-fn 6539 df-f 6540 |
This theorem is referenced by: (None) |
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