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Mirrors > Home > MPE Home > Th. List > tcphex | Structured version Visualization version GIF version |
Description: Lemma for tcphbas 25165 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
tcphex.v | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
tcphex | β’ (π₯ β π β¦ (ββ(π₯ , π₯))) β V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 β’ (π₯ β π β¦ (ββ(π₯ , π₯))) = (π₯ β π β¦ (ββ(π₯ , π₯))) | |
2 | fvrn0 6930 | . . . 4 β’ (ββ(π₯ , π₯)) β (ran β βͺ {β }) | |
3 | 2 | a1i 11 | . . 3 β’ (π₯ β π β (ββ(π₯ , π₯)) β (ran β βͺ {β })) |
4 | 1, 3 | fmpti 7125 | . 2 β’ (π₯ β π β¦ (ββ(π₯ , π₯))):πβΆ(ran β βͺ {β }) |
5 | tcphex.v | . . 3 β’ π = (Baseβπ) | |
6 | 5 | fvexi 6914 | . 2 β’ π β V |
7 | cnex 11225 | . . . 4 β’ β β V | |
8 | sqrtf 15348 | . . . . 5 β’ β:ββΆβ | |
9 | frn 6732 | . . . . 5 β’ (β:ββΆβ β ran β β β) | |
10 | 8, 9 | ax-mp 5 | . . . 4 β’ ran β β β |
11 | 7, 10 | ssexi 5324 | . . 3 β’ ran β β V |
12 | p0ex 5386 | . . 3 β’ {β } β V | |
13 | 11, 12 | unex 7752 | . 2 β’ (ran β βͺ {β }) β V |
14 | fex2 7945 | . 2 β’ (((π₯ β π β¦ (ββ(π₯ , π₯))):πβΆ(ran β βͺ {β }) β§ π β V β§ (ran β βͺ {β }) β V) β (π₯ β π β¦ (ββ(π₯ , π₯))) β V) | |
15 | 4, 6, 13, 14 | mp3an 1457 | 1 β’ (π₯ β π β¦ (ββ(π₯ , π₯))) β V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 Vcvv 3471 βͺ cun 3945 β wss 3947 β c0 4324 {csn 4630 β¦ cmpt 5233 ran crn 5681 βΆwf 6547 βcfv 6551 (class class class)co 7424 βcc 11142 βcsqrt 15218 Basecbs 17185 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 |
This theorem is referenced by: tcphbas 25165 tchplusg 25166 tcphmulr 25168 tcphsca 25169 tcphvsca 25170 tcphip 25171 tcphtopn 25172 tcphds 25177 rrxdim 33317 |
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