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Mirrors > Home > MPE Home > Th. List > restfn | Structured version Visualization version GIF version |
Description: The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
restfn | ⊢ ↾t Fn (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rest 16555 | . 2 ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | |
2 | vex 3418 | . . . 4 ⊢ 𝑗 ∈ V | |
3 | 2 | mptex 6814 | . . 3 ⊢ (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)) ∈ V |
4 | 3 | rnex 7434 | . 2 ⊢ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)) ∈ V |
5 | 1, 4 | fnmpoi 7578 | 1 ⊢ ↾t Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3415 ∩ cin 3830 ↦ cmpt 5009 × cxp 5406 ran crn 5409 Fn wfn 6185 ↾t crest 16553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-oprab 6982 df-mpo 6983 df-1st 7503 df-2nd 7504 df-rest 16555 |
This theorem is referenced by: 0rest 16562 restsspw 16564 firest 16565 restrcl 21472 restbas 21473 ssrest 21491 resstopn 21501 |
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