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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 6719 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
2 | 1 | inteqd 4873 | . 2 ⊢ (𝐹 Fn 𝐴 → ∩ ran 𝐹 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
3 | fvex 6677 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
4 | 3 | dfiin2 4951 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
5 | 2, 4 | syl6reqr 2875 | 1 ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {cab 2799 ∃wrex 3139 ∩ cint 4868 ∩ ciin 4912 ran crn 5550 Fn wfn 6344 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: firest 16700 pnrmopn 21945 txtube 22242 bcth3 23928 diaintclN 38188 dibintclN 38297 dihintcl 38474 imaiinfv 39283 |
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