Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evth2f | Structured version Visualization version GIF version |
Description: A version of evth2 23566 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
evth2f.1 | ⊢ Ⅎ𝑥𝐹 |
evth2f.2 | ⊢ Ⅎ𝑦𝐹 |
evth2f.3 | ⊢ Ⅎ𝑥𝑋 |
evth2f.4 | ⊢ Ⅎ𝑦𝑋 |
evth2f.5 | ⊢ 𝑋 = ∪ 𝐽 |
evth2f.6 | ⊢ 𝐾 = (topGen‘ran (,)) |
evth2f.7 | ⊢ (𝜑 → 𝐽 ∈ Comp) |
evth2f.8 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
evth2f.9 | ⊢ (𝜑 → 𝑋 ≠ ∅) |
Ref | Expression |
---|---|
evth2f | ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evth2f.5 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | evth2f.6 | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
3 | evth2f.7 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
4 | evth2f.8 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
5 | evth2f.9 | . . 3 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
6 | 1, 2, 3, 4, 5 | evth2 23566 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏)) |
7 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑎𝑋 | |
8 | evth2f.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
9 | evth2f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
10 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
11 | 9, 10 | nffv 6682 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑎) |
12 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
13 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
14 | 9, 13 | nffv 6682 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑏) |
15 | 11, 12, 14 | nfbr 5115 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑎) ≤ (𝐹‘𝑏) |
16 | 8, 15 | nfralw 3227 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) |
17 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑎∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) | |
18 | fveq2 6672 | . . . . . 6 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
19 | 18 | breq1d 5078 | . . . . 5 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
20 | 19 | ralbidv 3199 | . . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
21 | 7, 8, 16, 17, 20 | cbvrexfw 3440 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏)) |
22 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑏𝑋 | |
23 | evth2f.4 | . . . . 5 ⊢ Ⅎ𝑦𝑋 | |
24 | evth2f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
25 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑦𝑥 | |
26 | 24, 25 | nffv 6682 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
27 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑦 ≤ | |
28 | nfcv 2979 | . . . . . . 7 ⊢ Ⅎ𝑦𝑏 | |
29 | 24, 28 | nffv 6682 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑏) |
30 | 26, 27, 29 | nfbr 5115 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≤ (𝐹‘𝑏) |
31 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑏(𝐹‘𝑥) ≤ (𝐹‘𝑦) | |
32 | fveq2 6672 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) | |
33 | 32 | breq2d 5080 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
34 | 22, 23, 30, 31, 33 | cbvralfw 3439 | . . . 4 ⊢ (∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
35 | 34 | rexbii 3249 | . . 3 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
36 | 21, 35 | bitri 277 | . 2 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
37 | 6, 36 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ∅c0 4293 ∪ cuni 4840 class class class wbr 5068 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ≤ cle 10678 (,)cioo 12741 topGenctg 16713 Cn ccn 21834 Compccmp 21996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cn 21837 df-cnp 21838 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 |
This theorem is referenced by: stoweidlem29 42321 |
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