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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evthf | Structured version Visualization version GIF version |
Description: A version of evth 25004 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
evthf.1 | ⊢ Ⅎ𝑥𝐹 |
evthf.2 | ⊢ Ⅎ𝑦𝐹 |
evthf.3 | ⊢ Ⅎ𝑥𝑋 |
evthf.4 | ⊢ Ⅎ𝑦𝑋 |
evthf.5 | ⊢ Ⅎ𝑥𝜑 |
evthf.6 | ⊢ Ⅎ𝑦𝜑 |
evthf.7 | ⊢ 𝑋 = ∪ 𝐽 |
evthf.8 | ⊢ 𝐾 = (topGen‘ran (,)) |
evthf.9 | ⊢ (𝜑 → 𝐽 ∈ Comp) |
evthf.10 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
evthf.11 | ⊢ (𝜑 → 𝑋 ≠ ∅) |
Ref | Expression |
---|---|
evthf | ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evthf.7 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | evthf.8 | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
3 | evthf.9 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
4 | evthf.10 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
5 | evthf.11 | . . 3 ⊢ (𝜑 → 𝑋 ≠ ∅) | |
6 | 1, 2, 3, 4, 5 | evth 25004 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑏) ≤ (𝐹‘𝑎)) |
7 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑏𝑋 | |
8 | evthf.4 | . . . . 5 ⊢ Ⅎ𝑦𝑋 | |
9 | evthf.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
10 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑦𝑏 | |
11 | 9, 10 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑏) |
12 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑦 ≤ | |
13 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑦𝑎 | |
14 | 9, 13 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑎) |
15 | 11, 12, 14 | nfbr 5194 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑏) ≤ (𝐹‘𝑎) |
16 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑏(𝐹‘𝑦) ≤ (𝐹‘𝑎) | |
17 | fveq2 6906 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) | |
18 | 17 | breq1d 5157 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹‘𝑏) ≤ (𝐹‘𝑎) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝑎))) |
19 | 7, 8, 15, 16, 18 | cbvralfw 3301 | . . . 4 ⊢ (∀𝑏 ∈ 𝑋 (𝐹‘𝑏) ≤ (𝐹‘𝑎) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑎)) |
20 | 19 | rexbii 3091 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑏) ≤ (𝐹‘𝑎) ↔ ∃𝑎 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑎)) |
21 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑎𝑋 | |
22 | evthf.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
23 | evthf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
24 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
25 | 23, 24 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
26 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
27 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
28 | 23, 27 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑎) |
29 | 25, 26, 28 | nfbr 5194 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) ≤ (𝐹‘𝑎) |
30 | 22, 29 | nfralw 3308 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑎) |
31 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑎∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) | |
32 | fveq2 6906 | . . . . . 6 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
33 | 32 | breq2d 5159 | . . . . 5 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑦) ≤ (𝐹‘𝑎) ↔ (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
34 | 33 | ralbidv 3175 | . . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑎) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
35 | 21, 22, 30, 31, 34 | cbvrexfw 3302 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑎) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
36 | 20, 35 | bitri 275 | . 2 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑏) ≤ (𝐹‘𝑎) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
37 | 6, 36 | sylib 218 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∅c0 4338 ∪ cuni 4911 class class class wbr 5147 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ≤ cle 11293 (,)cioo 13383 topGenctg 17483 Cn ccn 23247 Compccmp 23409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 |
This theorem is referenced by: rfcnnnub 44973 |
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