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Mirrors > Home > MPE Home > Th. List > fniinfv | Structured version Visualization version GIF version |
Description: The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.) |
Ref | Expression |
---|---|
fniinfv | ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6658 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
2 | 1 | dfiin2 4921 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
3 | fnrnfv 6700 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
4 | 3 | inteqd 4843 | . 2 ⊢ (𝐹 Fn 𝐴 → ∩ ran 𝐹 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
5 | 2, 4 | eqtr4id 2852 | 1 ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 {cab 2776 ∃wrex 3107 ∩ cint 4838 ∩ ciin 4882 ran crn 5520 Fn wfn 6319 ‘cfv 6324 |
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-v 3443 df-sbc 3721 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-uni 4801 df-int 4839 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: firest 16698 pnrmopn 21948 txtube 22245 bcth3 23935 diaintclN 38354 dibintclN 38463 dihintcl 38640 imaiinfv 39634 |
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