imain - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > imain		Structured version Visualization version GIF version

Theorem imain 6583

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 6582	. . 3 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))))
2		imadif 6582	. . . 4 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
3	2	difeq2d 4066	. . 3 ⊢ (Fun ^◡𝐹 → ((𝐹 “ 𝐴) ∖ (𝐹 “ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
4	1, 3	eqtrd 2771	. 2 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))))
5		dfin4 4218	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵))
6	5	imaeq2i 6023	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐵)) = (𝐹 “ (𝐴 ∖ (𝐴 ∖ 𝐵)))
7		dfin4 4218	. 2 ⊢ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)) = ((𝐹 “ 𝐴) ∖ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
8	4, 6, 7	3eqtr4g 2796	1 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∖ cdif 3886 ∩ cin 3888 ^◡ccnv 5630 “ cima 5634 Fun wfun 6492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2728 df-clel 2811 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-br 5086 df-opab 5148 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-fun 6500

This theorem is referenced by: inpreima 7016 rnelfmlem 23917 fmfnfmlem3 23921 spthispth 29792 swrdrndisj 33017 ballotlemfrc 34671 poimirlem1 37942 poimirlem2 37943 poimirlem3 37944 poimirlem4 37945 poimirlem6 37947 poimirlem7 37948 poimirlem11 37952 poimirlem12 37953 poimirlem16 37957 poimirlem17 37958 poimirlem19 37960 poimirlem20 37961 poimirlem23 37964 poimirlem24 37965 poimirlem25 37966 poimirlem29 37970 poimirlem31 37972