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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 6391 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | imain 6409 | . 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 1538 ∩ cin 3880 ◡ccnv 5518 “ cima 5522 Fun wfun 6318 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 |
This theorem is referenced by: fimacnvinrn 6817 frnsuppeq 7825 ofco2 21056 cnrest2 21891 cnhaus 21959 kgencn3 22163 qtoptop2 22304 basqtop 22316 ismbfd 24243 mbfimaopn2 24261 i1fima 24282 i1fima2 24283 i1fd 24285 disjpreima 30347 sspreima 30406 smfco 43434 |
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