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Mirrors > Home > MPE Home > Th. List > fvco2 | Structured version Visualization version GIF version |
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.) |
Ref | Expression |
---|---|
fvco2 | ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsnfv 6258 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋})) | |
2 | 1 | imaeq2d 5466 | . . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋}))) |
3 | imaco 5640 | . . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋})) | |
4 | 2, 3 | syl6reqr 2675 | . . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)})) |
5 | 4 | eleq2d 2687 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
6 | 5 | iotabidv 5872 | . 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
7 | dffv3 6187 | . 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) | |
8 | dffv3 6187 | . 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})) | |
9 | 6, 7, 8 | 3eqtr4g 2681 | 1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 “ cima 5117 ∘ ccom 5118 ℩cio 5849 Fn wfn 5883 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: fvco 6274 fvco3 6275 fvco4i 6276 fvcofneq 6367 ofco 6917 curry1 7269 curry2 7272 enfixsn 8069 smobeth 9408 fpwwe 9468 addpqnq 9760 mulpqnq 9763 revco 13580 ccatco 13581 cshco 13582 swrdco 13583 isoval 16425 prdsidlem 17322 gsumwmhm 17382 prdsinvlem 17524 gsmsymgrfixlem1 17847 f1omvdconj 17866 pmtrfinv 17881 symggen 17890 symgtrinv 17892 pmtr3ncomlem1 17893 ringidval 18503 prdsmgp 18610 lmhmco 19043 evlslem1 19515 evlsvar 19523 chrrhm 19879 zrhcofipsgn 19939 dsmmbas2 20081 dsmm0cl 20084 frlmbas 20099 frlmup3 20139 frlmup4 20140 f1lindf 20161 lindfmm 20166 m1detdiag 20403 1stccnp 21265 prdstopn 21431 xpstopnlem2 21614 uniioombllem6 23356 0vfval 27461 cnre2csqlem 29956 mblfinlem2 33447 rabren3dioph 37379 hausgraph 37790 stoweidlem59 40276 afvco2 41256 |
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