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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 6251 | . . . . 5 ⊢ ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ (𝐺 “ {𝑋})) | |
2 | fnsnfv 6971 | . . . . . 6 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → {(𝐺‘𝑋)} = (𝐺 “ {𝑋})) | |
3 | 2 | imaeq2d 6060 | . . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 “ {(𝐺‘𝑋)}) = (𝐹 “ (𝐺 “ {𝑋}))) |
4 | 1, 3 | eqtr4id 2792 | . . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺) “ {𝑋}) = (𝐹 “ {(𝐺‘𝑋)})) |
5 | 4 | eleq2d 2820 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋}) ↔ 𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
6 | 5 | iotabidv 6528 | . 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)}))) |
7 | dffv3 6888 | . 2 ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (℩𝑥𝑥 ∈ ((𝐹 ∘ 𝐺) “ {𝑋})) | |
8 | dffv3 6888 | . 2 ⊢ (𝐹‘(𝐺‘𝑋)) = (℩𝑥𝑥 ∈ (𝐹 “ {(𝐺‘𝑋)})) | |
9 | 6, 7, 8 | 3eqtr4g 2798 | 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 4629 “ cima 5680 ∘ ccom 5681 ℩cio 6494 Fn wfn 6539 ‘cfv 6544 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 |
This theorem is referenced by: fvco 6990 fvco3 6991 fvco4i 6993 fvcofneq 7095 ofco 7693 curry1 8090 curry2 8093 fsplitfpar 8104 enfixsn 9081 updjudhcoinlf 9927 updjudhcoinrg 9928 updjud 9929 smobeth 10581 fpwwe 10641 addpqnq 10933 mulpqnq 10936 revco 14785 ccatco 14786 cshco 14787 swrdco 14788 isoval 17712 prdsidlem 18657 gsumwmhm 18726 prdsinvlem 18932 gsmsymgrfixlem1 19295 f1omvdconj 19314 pmtrfinv 19329 symggen 19338 symgtrinv 19340 pmtr3ncomlem1 19341 ringidval 20006 prdsmgp 20132 lmhmco 20654 chrrhm 21083 cofipsgn 21146 dsmmbas2 21292 dsmm0cl 21295 frlmbas 21310 frlmup3 21355 frlmup4 21356 f1lindf 21377 lindfmm 21382 evlslem1 21645 evlsvar 21653 m1detdiag 22099 1stccnp 22966 prdstopn 23132 xpstopnlem2 23315 uniioombllem6 25105 precsexlem1 27653 precsexlem2 27654 precsexlem3 27655 precsexlem4 27656 precsexlem5 27657 ex-fpar 29715 0vfval 29859 ghmquskerco 32529 cnre2csqlem 32890 mblfinlem2 36526 rabren3dioph 41553 hausgraph 41954 stoweidlem59 44775 afvco2 45884 isomushgr 46494 isomgrtrlem 46506 ackvalsucsucval 47374 |
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