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Mirrors > Home > MPE Home > Th. List > inpreima | Structured version Visualization version GIF version |
Description: Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.) |
Ref | Expression |
---|---|
inpreima | ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnvcnv 6167 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | imain 6185 | . 2 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∩ cin 3768 ◡ccnv 5311 “ cima 5315 Fun wfun 6095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-fun 6103 |
This theorem is referenced by: fimacnvinrn 6574 frnsuppeq 7544 ofco2 20583 cnrest2 21419 cnhaus 21487 kgencn3 21690 qtoptop2 21831 basqtop 21843 ismbfd 23747 mbfimaopn2 23765 i1fima 23786 i1fima2 23787 i1fd 23789 disjpreima 29914 sspreima 29966 smfco 41751 |
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