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Mirrors > Home > MPE Home > Th. List > axlowdimlem4 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28814. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem4.1 | β’ π΄ β β |
axlowdimlem4.2 | β’ π΅ β β |
Ref | Expression |
---|---|
axlowdimlem4 | β’ {β¨1, π΄β©, β¨2, π΅β©}:(1...2)βΆβ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ne2 12448 | . . . 4 β’ 1 β 2 | |
2 | 1ex 11238 | . . . . 5 β’ 1 β V | |
3 | 2ex 12317 | . . . . 5 β’ 2 β V | |
4 | axlowdimlem4.1 | . . . . . 6 β’ π΄ β β | |
5 | 4 | elexi 3484 | . . . . 5 β’ π΄ β V |
6 | axlowdimlem4.2 | . . . . . 6 β’ π΅ β β | |
7 | 6 | elexi 3484 | . . . . 5 β’ π΅ β V |
8 | 2, 3, 5, 7 | fpr 7158 | . . . 4 β’ (1 β 2 β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆ{π΄, π΅}) |
9 | 1, 8 | ax-mp 5 | . . 3 β’ {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆ{π΄, π΅} |
10 | fz12pr 13588 | . . . 4 β’ (1...2) = {1, 2} | |
11 | 10 | feq2i 6708 | . . 3 β’ ({β¨1, π΄β©, β¨2, π΅β©}:(1...2)βΆ{π΄, π΅} β {β¨1, π΄β©, β¨2, π΅β©}:{1, 2}βΆ{π΄, π΅}) |
12 | 9, 11 | mpbir 230 | . 2 β’ {β¨1, π΄β©, β¨2, π΅β©}:(1...2)βΆ{π΄, π΅} |
13 | 4, 6 | pm3.2i 469 | . . 3 β’ (π΄ β β β§ π΅ β β) |
14 | 5, 7 | prss 4819 | . . 3 β’ ((π΄ β β β§ π΅ β β) β {π΄, π΅} β β) |
15 | 13, 14 | mpbi 229 | . 2 β’ {π΄, π΅} β β |
16 | fss 6733 | . 2 β’ (({β¨1, π΄β©, β¨2, π΅β©}:(1...2)βΆ{π΄, π΅} β§ {π΄, π΅} β β) β {β¨1, π΄β©, β¨2, π΅β©}:(1...2)βΆβ) | |
17 | 12, 15, 16 | mp2an 690 | 1 β’ {β¨1, π΄β©, β¨2, π΅β©}:(1...2)βΆβ |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 394 β wcel 2098 β wne 2930 β wss 3940 {cpr 4626 β¨cop 4630 βΆwf 6538 (class class class)co 7415 βcr 11135 1c1 11137 2c2 12295 ...cfz 13514 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 |
This theorem is referenced by: axlowdimlem5 28799 axlowdimlem6 28800 axlowdimlem17 28811 |
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