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Theorem imain 6621

Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

**Assertion**
Ref	Expression
imain	⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

**Proof of Theorem imain**
Step	Hyp	Ref	Expression
1		imadif 6620	. . 3 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))))
2		imadif 6620	. . . 4 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
3	2	difeq2d 4081	. . 3 ⊢ (Fun ^◡𝐹 → ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
4	1, 3	eqtrd 2798	. 2 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
5		dfin4 4231	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	imaeq2i 6060	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵)))
7		dfin4 4231	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
8	4, 6, 7	3eqtr4g 2823	1 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1570 ∖ cdif 3902 ∩ cin 3904 ^◡ccnv 5660 “ cima 5664 Fun wfun 6530

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-clab 2742 df-cleq 2755 df-clel 2838 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-br 5110 df-opab 5174 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-fun 6538

This theorem is referenced by: inpreima 7059 rnelfmlem 24109 fmfnfmlem3 24113 spthispth 30073 swrdrndisj 33277 ballotlemfrc 34917 poimirlem1 38272 poimirlem2 38273 poimirlem3 38274 poimirlem4 38275 poimirlem6 38277 poimirlem7 38278 poimirlem11 38282 poimirlem12 38283 poimirlem16 38287 poimirlem17 38288 poimirlem19 38290 poimirlem20 38291 poimirlem23 38294 poimirlem24 38295 poimirlem25 38296 poimirlem29 38300 poimirlem31 38302