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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 | fnrnfv 6468 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
2 | 1 | inteqd 4673 | . 2 ⊢ (𝐹 Fn 𝐴 → ∩ ran 𝐹 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
3 | fvex 6425 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
4 | 3 | dfiin2 4746 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
5 | 2, 4 | syl6reqr 2853 | 1 ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 {cab 2786 ∃wrex 3091 ∩ cint 4668 ∩ ciin 4712 ran crn 5314 Fn wfn 6097 ‘cfv 6102 |
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 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
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 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-int 4669 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-iota 6065 df-fun 6104 df-fn 6105 df-fv 6110 |
This theorem is referenced by: firest 16407 pnrmopn 21475 txtube 21771 bcth3 23456 diaintclN 37078 dibintclN 37187 dihintcl 37364 imaiinfv 38037 |
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