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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 23167 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 23167 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏)) |
7 | nfcv 2934 | . . . 4 ⊢ Ⅎ𝑎𝑋 | |
8 | evth2f.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
9 | evth2f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
10 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
11 | 9, 10 | nffv 6456 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑎) |
12 | nfcv 2934 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
13 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
14 | 9, 13 | nffv 6456 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑏) |
15 | 11, 12, 14 | nfbr 4933 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑎) ≤ (𝐹‘𝑏) |
16 | 8, 15 | nfral 3127 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) |
17 | nfv 1957 | . . . 4 ⊢ Ⅎ𝑎∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) | |
18 | fveq2 6446 | . . . . . 6 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
19 | 18 | breq1d 4896 | . . . . 5 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
20 | 19 | ralbidv 3168 | . . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
21 | 7, 8, 16, 17, 20 | cbvrexf 3362 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏)) |
22 | nfcv 2934 | . . . . 5 ⊢ Ⅎ𝑏𝑋 | |
23 | evth2f.4 | . . . . 5 ⊢ Ⅎ𝑦𝑋 | |
24 | evth2f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
25 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑦𝑥 | |
26 | 24, 25 | nffv 6456 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
27 | nfcv 2934 | . . . . . 6 ⊢ Ⅎ𝑦 ≤ | |
28 | nfcv 2934 | . . . . . . 7 ⊢ Ⅎ𝑦𝑏 | |
29 | 24, 28 | nffv 6456 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑏) |
30 | 26, 27, 29 | nfbr 4933 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≤ (𝐹‘𝑏) |
31 | nfv 1957 | . . . . 5 ⊢ Ⅎ𝑏(𝐹‘𝑥) ≤ (𝐹‘𝑦) | |
32 | fveq2 6446 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) | |
33 | 32 | breq2d 4898 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
34 | 22, 23, 30, 31, 33 | cbvralf 3361 | . . . 4 ⊢ (∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
35 | 34 | rexbii 3224 | . . 3 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
36 | 21, 35 | bitri 267 | . 2 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
37 | 6, 36 | sylib 210 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Ⅎwnfc 2919 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 ∅c0 4141 ∪ cuni 4671 class class class wbr 4886 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ≤ cle 10412 (,)cioo 12487 topGenctg 16484 Cn ccn 21436 Compccmp 21598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cn 21439 df-cnp 21440 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-xms 22533 df-ms 22534 df-tms 22535 |
This theorem is referenced by: stoweidlem29 41173 |
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