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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 6915 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
2 | 1 | dfiin2 5041 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
3 | fnrnfv 6963 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
4 | 3 | inteqd 4958 | . 2 ⊢ (𝐹 Fn 𝐴 → ∩ ran 𝐹 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
5 | 2, 4 | eqtr4id 2787 | 1 ⊢ (𝐹 Fn 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {cab 2705 ∃wrex 3067 ∩ cint 4953 ∩ ciin 5001 ran crn 5683 Fn wfn 6548 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 |
This theorem is referenced by: firest 17421 pnrmopn 23267 txtube 23564 bcth3 25279 diaintclN 40563 dibintclN 40672 dihintcl 40849 imaiinfv 42144 |
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