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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 6633 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
| 2 | imain 6651 | . 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 1540 ∩ cin 3950 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: cnvimainrn 7087 sspreima 7088 fimacnvinrn 7091 fsuppeq 8200 fsuppeqg 8201 ofco2 22457 cnrest2 23294 cnhaus 23362 kgencn3 23566 qtoptop2 23707 basqtop 23719 ismbfd 25674 mbfimaopn2 25692 i1fima 25713 i1fima2 25714 i1fd 25716 disjpreima 32597 smfco 46817 predisj 48730 |
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