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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 6071 | . . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋})) | |
2 | fnsnfv 6718 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋})) | |
3 | 2 | imaeq2d 5896 | . . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋}))) |
4 | 1, 3 | eqtr4id 2852 | . . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)})) |
5 | 4 | eleq2d 2875 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
6 | 5 | iotabidv 6308 | . 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
7 | dffv3 6641 | . 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) | |
8 | dffv3 6641 | . 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})) | |
9 | 6, 7, 8 | 3eqtr4g 2858 | 1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 “ cima 5522 ∘ ccom 5523 ℩cio 6281 Fn wfn 6319 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: fvco 6736 fvco3 6737 fvco4i 6739 fvcofneq 6836 ofco 7409 curry1 7782 curry2 7785 fsplitfpar 7797 enfixsn 8609 updjudhcoinlf 9345 updjudhcoinrg 9346 updjud 9347 smobeth 9997 fpwwe 10057 addpqnq 10349 mulpqnq 10352 revco 14187 ccatco 14188 cshco 14189 swrdco 14190 isoval 17027 prdsidlem 17935 gsumwmhm 18002 prdsinvlem 18200 gsmsymgrfixlem1 18547 f1omvdconj 18566 pmtrfinv 18581 symggen 18590 symgtrinv 18592 pmtr3ncomlem1 18593 ringidval 19246 prdsmgp 19356 lmhmco 19808 chrrhm 20223 cofipsgn 20282 dsmmbas2 20426 dsmm0cl 20429 frlmbas 20444 frlmup3 20489 frlmup4 20490 f1lindf 20511 lindfmm 20516 evlslem1 20754 evlsvar 20762 m1detdiag 21202 1stccnp 22067 prdstopn 22233 xpstopnlem2 22416 uniioombllem6 24192 ex-fpar 28247 0vfval 28389 cnre2csqlem 31263 mblfinlem2 35095 rabren3dioph 39756 hausgraph 40156 stoweidlem59 42701 afvco2 43732 isomushgr 44344 isomgrtrlem 44356 ackvalsucsucval 45102 |
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