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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 24906 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 24906 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏)) |
7 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑎𝑋 | |
8 | evth2f.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
9 | evth2f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
10 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
11 | 9, 10 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑎) |
12 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
13 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
14 | 9, 13 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑏) |
15 | 11, 12, 14 | nfbr 5199 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑎) ≤ (𝐹‘𝑏) |
16 | 8, 15 | nfralw 3306 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) |
17 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑎∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) | |
18 | fveq2 6902 | . . . . . 6 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
19 | 18 | breq1d 5162 | . . . . 5 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
20 | 19 | ralbidv 3175 | . . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
21 | 7, 8, 16, 17, 20 | cbvrexfw 3300 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏)) |
22 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑏𝑋 | |
23 | evth2f.4 | . . . . 5 ⊢ Ⅎ𝑦𝑋 | |
24 | evth2f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
25 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑦𝑥 | |
26 | 24, 25 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
27 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑦 ≤ | |
28 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑦𝑏 | |
29 | 24, 28 | nffv 6912 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑏) |
30 | 26, 27, 29 | nfbr 5199 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≤ (𝐹‘𝑏) |
31 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑏(𝐹‘𝑥) ≤ (𝐹‘𝑦) | |
32 | fveq2 6902 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) | |
33 | 32 | breq2d 5164 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
34 | 22, 23, 30, 31, 33 | cbvralfw 3299 | . . . 4 ⊢ (∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
35 | 34 | rexbii 3091 | . . 3 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
36 | 21, 35 | bitri 274 | . 2 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
37 | 6, 36 | sylib 217 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2879 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∅c0 4326 ∪ cuni 4912 class class class wbr 5152 ran crn 5683 ‘cfv 6553 (class class class)co 7426 ≤ cle 11287 (,)cioo 13364 topGenctg 17426 Cn ccn 23148 Compccmp 23310 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-icc 13371 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cn 23151 df-cnp 23152 df-cmp 23311 df-tx 23486 df-hmeo 23679 df-xms 24246 df-ms 24247 df-tms 24248 |
This theorem is referenced by: stoweidlem29 45446 |
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