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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 4078	. . 3 ⊢ (Fun ^◡𝐹 → ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
4	1, 3	eqtrd 2796	. 2 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
5		dfin4 4228	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	imaeq2i 6043	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵)))
7		dfin4 4228	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
8	4, 6, 7	3eqtr4g 2821	1 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∖ cdif 3899 ∩ cin 3901 ^◡ccnv 5642 “ cima 5646 Fun wfun 6510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-pr 5387

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-clab 2740 df-cleq 2753 df-clel 2836 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-sn 4580 df-pr 4582 df-op 4586 df-br 5098 df-opab 5160 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-fun 6518

This theorem is referenced by: inpreima 7040 rnelfmlem 24000 fmfnfmlem3 24004 spthispth 29881 swrdrndisj 33096 ballotlemfrc 34785 poimirlem1 38081 poimirlem2 38082 poimirlem3 38083 poimirlem4 38084 poimirlem6 38086 poimirlem7 38087 poimirlem11 38091 poimirlem12 38092 poimirlem16 38096 poimirlem17 38097 poimirlem19 38099 poimirlem20 38100 poimirlem23 38103 poimirlem24 38104 poimirlem25 38105 poimirlem29 38109 poimirlem31 38111