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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 17367 | . 2 ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | |
2 | vex 3470 | . . . 4 ⊢ 𝑗 ∈ V | |
3 | 2 | mptex 7216 | . . 3 ⊢ (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)) ∈ V |
4 | 3 | rnex 7896 | . 2 ⊢ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)) ∈ V |
5 | 1, 4 | fnmpoi 8049 | 1 ⊢ ↾t Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3466 ∩ cin 3939 ↦ cmpt 5221 × cxp 5664 ran crn 5667 Fn wfn 6528 ↾t crest 17365 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-rest 17367 |
This theorem is referenced by: 0rest 17374 restsspw 17376 firest 17377 restrcl 22983 restbas 22984 ssrest 23002 resstopn 23012 |
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