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Theorem imain 6601

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 6600	. . 3 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))))
2		imadif 6600	. . . 4 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
3	2	difeq2d 4089	. . 3 ⊢ (Fun ^◡𝐹 → ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
4	1, 3	eqtrd 2764	. 2 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
5		dfin4 4241	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	imaeq2i 6029	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵)))
7		dfin4 4241	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
8	4, 6, 7	3eqtr4g 2789	1 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∖ cdif 3911 ∩ cin 3913 ^◡ccnv 5637 “ cima 5641 Fun wfun 6505

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-br 5108 df-opab 5170 df-id 5533 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-fun 6513

This theorem is referenced by: inpreima 7036 rnelfmlem 23839 fmfnfmlem3 23843 spthispth 29654 swrdrndisj 32879 ballotlemfrc 34518 poimirlem1 37615 poimirlem2 37616 poimirlem3 37617 poimirlem4 37618 poimirlem6 37620 poimirlem7 37621 poimirlem11 37625 poimirlem12 37626 poimirlem16 37630 poimirlem17 37631 poimirlem19 37633 poimirlem20 37634 poimirlem23 37637 poimirlem24 37638 poimirlem25 37639 poimirlem29 37643 poimirlem31 37645