![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > evth2f | Structured version Visualization version GIF version |
Description: A version of evth2 24988 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 24988 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏)) |
7 | nfcv 2901 | . . . 4 ⊢ Ⅎ𝑎𝑋 | |
8 | evth2f.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
9 | evth2f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
10 | nfcv 2901 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
11 | 9, 10 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑎) |
12 | nfcv 2901 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
13 | nfcv 2901 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
14 | 9, 13 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑏) |
15 | 11, 12, 14 | nfbr 5197 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑎) ≤ (𝐹‘𝑏) |
16 | 8, 15 | nfralw 3307 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) |
17 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑎∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) | |
18 | fveq2 6902 | . . . . . 6 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
19 | 18 | breq1d 5160 | . . . . 5 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
20 | 19 | ralbidv 3174 | . . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
21 | 7, 8, 16, 17, 20 | cbvrexfw 3301 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏)) |
22 | nfcv 2901 | . . . . 5 ⊢ Ⅎ𝑏𝑋 | |
23 | evth2f.4 | . . . . 5 ⊢ Ⅎ𝑦𝑋 | |
24 | evth2f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
25 | nfcv 2901 | . . . . . . 7 ⊢ Ⅎ𝑦𝑥 | |
26 | 24, 25 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
27 | nfcv 2901 | . . . . . 6 ⊢ Ⅎ𝑦 ≤ | |
28 | nfcv 2901 | . . . . . . 7 ⊢ Ⅎ𝑦𝑏 | |
29 | 24, 28 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑏) |
30 | 26, 27, 29 | nfbr 5197 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≤ (𝐹‘𝑏) |
31 | nfv 1910 | . . . . 5 ⊢ Ⅎ𝑏(𝐹‘𝑥) ≤ (𝐹‘𝑦) | |
32 | fveq2 6902 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) | |
33 | 32 | breq2d 5162 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
34 | 22, 23, 30, 31, 33 | cbvralfw 3300 | . . . 4 ⊢ (∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
35 | 34 | rexbii 3090 | . . 3 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
36 | 21, 35 | bitri 275 | . 2 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
37 | 6, 36 | sylib 218 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 Ⅎwnfc 2886 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 ∅c0 4339 ∪ cuni 4915 class class class wbr 5150 ran crn 5685 ‘cfv 6559 (class class class)co 7426 ≤ cle 11288 (,)cioo 13378 topGenctg 17474 Cn ccn 23230 Compccmp 23392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7748 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6318 df-ord 6384 df-on 6385 df-lim 6386 df-suc 6387 df-iota 6511 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7882 df-1st 8008 df-2nd 8009 df-supp 8180 df-frecs 8300 df-wrecs 8331 df-recs 8405 df-rdg 8444 df-1o 8500 df-2o 8501 df-er 8739 df-map 8862 df-ixp 8932 df-en 8980 df-dom 8981 df-sdom 8982 df-fin 8983 df-fsupp 9395 df-fi 9443 df-sup 9474 df-inf 9475 df-oi 9542 df-card 9971 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11486 df-neg 11487 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12606 df-dec 12726 df-uz 12871 df-q 12983 df-rp 13027 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-icc 13385 df-fz 13539 df-fzo 13683 df-seq 14030 df-exp 14090 df-hash 14357 df-cj 15125 df-re 15126 df-im 15127 df-sqrt 15261 df-abs 15262 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17236 df-ress 17265 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17459 df-topn 17460 df-0g 17478 df-gsum 17479 df-topgen 17480 df-pt 17481 df-prds 17484 df-xrs 17539 df-qtop 17544 df-imas 17545 df-xps 17547 df-mre 17621 df-mrc 17622 df-acs 17624 df-mgm 18655 df-sgrp 18734 df-mnd 18750 df-submnd 18796 df-mulg 19085 df-cntz 19334 df-cmn 19801 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22898 df-topon 22915 df-topsp 22937 df-bases 22951 df-cn 23233 df-cnp 23234 df-cmp 23393 df-tx 23568 df-hmeo 23761 df-xms 24328 df-ms 24329 df-tms 24330 |
This theorem is referenced by: stoweidlem29 45942 |
Copyright terms: Public domain | W3C validator |