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Mirrors > Home > MPE Home > Th. List > fcoi1 | Structured version Visualization version GIF version |
Description: Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fcoi1 | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6669 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | df-fn 6500 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
3 | eqimss 4001 | . . . . 5 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | cnvi 6095 | . . . . . . . . . 10 ⊢ ◡ I = I | |
5 | 4 | reseq1i 5934 | . . . . . . . . 9 ⊢ (◡ I ↾ 𝐴) = ( I ↾ 𝐴) |
6 | 5 | cnveqi 5831 | . . . . . . . 8 ⊢ ◡(◡ I ↾ 𝐴) = ◡( I ↾ 𝐴) |
7 | cnvresid 6581 | . . . . . . . 8 ⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | |
8 | 6, 7 | eqtr2i 2766 | . . . . . . 7 ⊢ ( I ↾ 𝐴) = ◡(◡ I ↾ 𝐴) |
9 | 8 | coeq2i 5817 | . . . . . 6 ⊢ (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ ◡(◡ I ↾ 𝐴)) |
10 | cores2 6212 | . . . . . 6 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ◡(◡ I ↾ 𝐴)) = (𝐹 ∘ I )) | |
11 | 9, 10 | eqtrid 2789 | . . . . 5 ⊢ (dom 𝐹 ⊆ 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I )) |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = (𝐹 ∘ I )) |
13 | funrel 6519 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
14 | coi1 6215 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹) |
16 | 12, 15 | sylan9eqr 2799 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
17 | 2, 16 | sylbi 216 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
18 | 1, 17 | syl 17 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ⊆ wss 3911 I cid 5531 ◡ccnv 5633 dom cdm 5634 ↾ cres 5636 ∘ ccom 5638 Rel wrel 5639 Fun wfun 6491 Fn wfn 6492 ⟶wf 6493 |
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-10 2138 ax-12 2172 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-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 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-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: fcof1oinvd 7240 mapen 9086 mapfien 9345 hashfacen 14352 hashfacenOLD 14353 cofurid 17778 setccatid 17971 estrccatid 18020 curf2ndf 18137 efmndid 18699 efmndmnd 18700 f1omvdco2 19231 psgnunilem1 19276 pf1mpf 21721 pf1ind 21724 wilthlem3 26422 hoico1 30701 fmptco1f1o 31550 fcobijfs 31643 cycpmconjslem2 32007 cycpmconjs 32008 cyc3conja 32009 reprpmtf1o 33242 ltrncoidN 38594 trlcoabs2N 39188 trlcoat 39189 cdlemg47a 39200 cdlemg46 39201 trljco 39206 tendo1mulr 39237 tendo0co2 39254 cdlemi2 39285 cdlemk2 39298 cdlemk4 39300 cdlemk8 39304 cdlemk53 39423 cdlemk55a 39425 dvhopN 39582 dihopelvalcpre 39714 dihmeetlem1N 39756 dihglblem5apreN 39757 diophrw 41085 mendring 41522 rngccatidALTV 46294 ringccatidALTV 46357 |
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