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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 6490 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | imain 6508 | . 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 1539 ∩ cin 3887 ◡ccnv 5584 “ cima 5588 Fun wfun 6417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3429 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-br 5076 df-opab 5138 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-fun 6425 |
This theorem is referenced by: cnvimainrn 6931 sspreima 6932 fimacnvinrn 6936 frnsuppeq 7967 frnsuppeqg 7968 ofco2 21544 cnrest2 22381 cnhaus 22449 kgencn3 22653 qtoptop2 22794 basqtop 22806 ismbfd 24746 mbfimaopn2 24764 i1fima 24785 i1fima2 24786 i1fd 24788 disjpreima 30864 smfco 44265 predisj 46086 |
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