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Mirrors > Home > MPE Home > Th. List > infxpenc2lem2OLD | Unicode version |
Description: Lemma for infxpenc2OLD 8424. (Contributed by Mario Carneiro, 30-May-2015.) Obsolete version of infxpenc2lem2 8418 as of 7-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
infxpenc2lem2OLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxpenc2OLD.1 | . . 3 | |
2 | mptexg 6142 | . . 3 | |
3 | 1, 2 | syl 16 | . 2 |
4 | 1 | adantr 465 | . . . . . . 7 |
5 | simprl 756 | . . . . . . 7 | |
6 | onelon 4908 | . . . . . . 7 | |
7 | 4, 5, 6 | syl2anc 661 | . . . . . 6 |
8 | simprr 757 | . . . . . 6 | |
9 | infxpenc2OLD.2 | . . . . . . . 8 | |
10 | infxpenc2OLD.3 | . . . . . . . 8 | |
11 | 1, 9, 10 | infxpenc2lem1 8417 | . . . . . . 7 |
12 | 11 | simpld 459 | . . . . . 6 |
13 | infxpenc2OLD.4 | . . . . . . 7 | |
14 | 13 | adantr 465 | . . . . . 6 |
15 | infxpenc2OLD.5 | . . . . . . 7 | |
16 | 15 | adantr 465 | . . . . . 6 |
17 | 11 | simprd 463 | . . . . . 6 |
18 | infxpenc2OLD.k | . . . . . 6 | |
19 | infxpenc2OLD.h | . . . . . 6 | |
20 | infxpenc2OLD.l | . . . . . 6 | |
21 | infxpenc2OLD.x | . . . . . 6 | |
22 | infxpenc2OLD.y | . . . . . 6 | |
23 | infxpenc2OLD.j | . . . . . 6 | |
24 | infxpenc2OLD.z | . . . . . 6 | |
25 | infxpenc2OLD.t | . . . . . 6 | |
26 | infxpenc2OLD.g | . . . . . 6 | |
27 | 7, 8, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 | infxpencOLD 8421 | . . . . 5 |
28 | f1of 5821 | . . . . . . . . 9 | |
29 | 27, 28 | syl 16 | . . . . . . . 8 |
30 | vex 3112 | . . . . . . . . 9 | |
31 | 30, 30 | xpex 6604 | . . . . . . . 8 |
32 | fex 6145 | . . . . . . . 8 | |
33 | 29, 31, 32 | sylancl 662 | . . . . . . 7 |
34 | eqid 2457 | . . . . . . . 8 | |
35 | 34 | fvmpt2 5963 | . . . . . . 7 |
36 | 5, 33, 35 | syl2anc 661 | . . . . . 6 |
37 | f1oeq1 5812 | . . . . . 6 | |
38 | 36, 37 | syl 16 | . . . . 5 |
39 | 27, 38 | mpbird 232 | . . . 4 |
40 | 39 | expr 615 | . . 3 |
41 | 40 | ralrimiva 2871 | . 2 |
42 | nfmpt1 4541 | . . . . 5 | |
43 | 42 | nfeq2 2636 | . . . 4 |
44 | fveq1 5870 | . . . . . 6 | |
45 | f1oeq1 5812 | . . . . . 6 | |
46 | 44, 45 | syl 16 | . . . . 5 |
47 | 46 | imbi2d 316 | . . . 4 |
48 | 43, 47 | ralbid 2891 | . . 3 |
49 | 48 | spcegv 3195 | . 2 |
50 | 3, 41, 49 | sylc 60 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
/\ wa 369 = wceq 1395 E. wex 1612
e. wcel 1818 A. wral 2807 E. wrex 2808
{ crab 2811 cvv 3109
\ cdif 3472 C_ wss 3475 c0 3784 <. cop 4035 e. cmpt 4510
cid 4795
con0 4883 X. cxp 5002 `' ccnv 5003
ran crn 5005 |` cres 5006 " cima 5007
o. ccom 5008 --> wf 5589 -1-1-onto-> wf1o 5592 ` cfv 5593 (class class class)co 6296
e. cmpt2 6298 com 6700
c1o 7142
c2o 7143
coa 7146
comu 7147
coe 7148
cmap 7439
cfn 7536 ccnf 8099 |
This theorem is referenced by: infxpenc2lem3OLD 8423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-inf2 8079 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-fal 1401 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-int 4287 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-supp 6919 df-recs 7061 df-rdg 7095 df-seqom 7132 df-1o 7149 df-2o 7150 df-oadd 7153 df-omul 7154 df-oexp 7155 df-er 7330 df-map 7441 df-en 7537 df-dom 7538 df-sdom 7539 df-fin 7540 df-fsupp 7850 df-oi 7956 df-cnf 8100 |
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