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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 | imaco 6208 | . . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋})) | |
2 | fnsnfv 6925 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋})) | |
3 | 2 | imaeq2d 6018 | . . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋}))) |
4 | 1, 3 | eqtr4id 2796 | . . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)})) |
5 | 4 | eleq2d 2824 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
6 | 5 | iotabidv 6485 | . 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
7 | dffv3 6843 | . 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) | |
8 | dffv3 6843 | . 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})) | |
9 | 6, 7, 8 | 3eqtr4g 2802 | 1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4591 “ cima 5641 ∘ ccom 5642 ℩cio 6451 Fn wfn 6496 ‘cfv 6501 |
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-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-fv 6509 |
This theorem is referenced by: fvco 6944 fvco3 6945 fvco4i 6947 fvcofneq 7048 ofco 7645 curry1 8041 curry2 8044 fsplitfpar 8055 enfixsn 9032 updjudhcoinlf 9875 updjudhcoinrg 9876 updjud 9877 smobeth 10529 fpwwe 10589 addpqnq 10881 mulpqnq 10884 revco 14730 ccatco 14731 cshco 14732 swrdco 14733 isoval 17655 prdsidlem 18595 gsumwmhm 18662 prdsinvlem 18863 gsmsymgrfixlem1 19216 f1omvdconj 19235 pmtrfinv 19250 symggen 19259 symgtrinv 19261 pmtr3ncomlem1 19262 ringidval 19922 prdsmgp 20041 lmhmco 20520 chrrhm 20950 cofipsgn 21013 dsmmbas2 21159 dsmm0cl 21162 frlmbas 21177 frlmup3 21222 frlmup4 21223 f1lindf 21244 lindfmm 21249 evlslem1 21508 evlsvar 21516 m1detdiag 21962 1stccnp 22829 prdstopn 22995 xpstopnlem2 23178 uniioombllem6 24968 ex-fpar 29448 0vfval 29590 ghmquskerco 32236 cnre2csqlem 32531 mblfinlem2 36145 rabren3dioph 41167 hausgraph 41568 stoweidlem59 44374 afvco2 45482 isomushgr 46092 isomgrtrlem 46104 ackvalsucsucval 46848 |
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