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Theorem imain 6536

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 6535	. . 3 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))))
2		imadif 6535	. . . 4 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
3	2	difeq2d 4060	. . 3 ⊢ (Fun ^◡𝐹 → ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
4	1, 3	eqtrd 2773	. 2 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
5		dfin4 4204	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	imaeq2i 5968	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵)))
7		dfin4 4204	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
8	4, 6, 7	3eqtr4g 2798	1 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∖ cdif 3886 ∩ cin 3888 ^◡ccnv 5590 “ cima 5594 Fun wfun 6441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-br 5078 df-opab 5140 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-fun 6449

This theorem is referenced by: inpreima 6961 rnelfmlem 23131 fmfnfmlem3 23135 spthispth 28122 swrdrndisj 31257 ballotlemfrc 32521 poimirlem1 35806 poimirlem2 35807 poimirlem3 35808 poimirlem4 35809 poimirlem6 35811 poimirlem7 35812 poimirlem11 35816 poimirlem12 35817 poimirlem16 35821 poimirlem17 35822 poimirlem19 35824 poimirlem20 35825 poimirlem23 35828 poimirlem24 35829 poimirlem25 35830 poimirlem29 35834 poimirlem31 35836