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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 6746 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋})) | |
2 | 1 | imaeq2d 5932 | . . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋}))) |
3 | imaco 6107 | . . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋})) | |
4 | 2, 3 | syl6reqr 2878 | . . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)})) |
5 | 4 | eleq2d 2901 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
6 | 5 | iotabidv 6342 | . 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
7 | dffv3 6669 | . 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) | |
8 | dffv3 6669 | . 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})) | |
9 | 6, 7, 8 | 3eqtr4g 2884 | 1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {csn 4570 “ cima 5561 ∘ ccom 5562 ℩cio 6315 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: fvco 6762 fvco3 6763 fvco4i 6765 fvcofneq 6862 ofco 7432 curry1 7802 curry2 7805 fsplitfpar 7817 enfixsn 8629 updjudhcoinlf 9364 updjudhcoinrg 9365 updjud 9366 smobeth 10011 fpwwe 10071 addpqnq 10363 mulpqnq 10366 revco 14199 ccatco 14200 cshco 14201 swrdco 14202 isoval 17038 prdsidlem 17946 gsumwmhm 18013 prdsinvlem 18211 gsmsymgrfixlem1 18558 f1omvdconj 18577 pmtrfinv 18592 symggen 18601 symgtrinv 18603 pmtr3ncomlem1 18604 ringidval 19256 prdsmgp 19363 lmhmco 19818 evlslem1 20298 evlsvar 20306 chrrhm 20681 cofipsgn 20740 dsmmbas2 20884 dsmm0cl 20887 frlmbas 20902 frlmup3 20947 frlmup4 20948 f1lindf 20969 lindfmm 20974 m1detdiag 21209 1stccnp 22073 prdstopn 22239 xpstopnlem2 22422 uniioombllem6 24192 ex-fpar 28244 0vfval 28386 cnre2csqlem 31157 mblfinlem2 34934 rabren3dioph 39418 hausgraph 39818 stoweidlem59 42351 afvco2 43382 isomushgr 43998 isomgrtrlem 44010 |
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