| 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 24982 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 24982 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏)) |
| 7 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑎𝑋 | |
| 8 | evth2f.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
| 9 | evth2f.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
| 10 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥𝑎 | |
| 11 | 9, 10 | nffv 6914 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑎) |
| 12 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
| 13 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥𝑏 | |
| 14 | 9, 13 | nffv 6914 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑏) |
| 15 | 11, 12, 14 | nfbr 5188 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑎) ≤ (𝐹‘𝑏) |
| 16 | 8, 15 | nfralw 3310 | . . . 4 ⊢ Ⅎ𝑥∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) |
| 17 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑎∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) | |
| 18 | fveq2 6904 | . . . . . 6 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
| 19 | 18 | breq1d 5151 | . . . . 5 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
| 20 | 19 | ralbidv 3177 | . . . 4 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏))) |
| 21 | 7, 8, 16, 17, 20 | cbvrexfw 3304 | . . 3 ⊢ (∃𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑎) ≤ (𝐹‘𝑏) ↔ ∃𝑥 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏)) |
| 22 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑏𝑋 | |
| 23 | evth2f.4 | . . . . 5 ⊢ Ⅎ𝑦𝑋 | |
| 24 | evth2f.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐹 | |
| 25 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑦𝑥 | |
| 26 | 24, 25 | nffv 6914 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑥) |
| 27 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑦 ≤ | |
| 28 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑦𝑏 | |
| 29 | 24, 28 | nffv 6914 | . . . . . 6 ⊢ Ⅎ𝑦(𝐹‘𝑏) |
| 30 | 26, 27, 29 | nfbr 5188 | . . . . 5 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≤ (𝐹‘𝑏) |
| 31 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑏(𝐹‘𝑥) ≤ (𝐹‘𝑦) | |
| 32 | fveq2 6904 | . . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐹‘𝑏) = (𝐹‘𝑦)) | |
| 33 | 32 | breq2d 5153 | . . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| 34 | 22, 23, 30, 31, 33 | cbvralfw 3303 | . . . 4 ⊢ (∀𝑏 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑏) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
| 35 | 34 | rexbii 3093 | . . 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 1540 ∈ wcel 2108 Ⅎwnfc 2889 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∅c0 4332 ∪ cuni 4905 class class class wbr 5141 ran crn 5684 ‘cfv 6559 (class class class)co 7429 ≤ cle 11292 (,)cioo 13383 topGenctg 17478 Cn ccn 23222 Compccmp 23384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 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 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-q 12987 df-rp 13031 df-xneg 13150 df-xadd 13151 df-xmul 13152 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 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-hom 17317 df-cco 17318 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17543 df-qtop 17548 df-imas 17549 df-xps 17551 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-mulg 19082 df-cntz 19331 df-cmn 19796 df-psmet 21348 df-xmet 21349 df-met 21350 df-bl 21351 df-mopn 21352 df-cnfld 21357 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cn 23225 df-cnp 23226 df-cmp 23385 df-tx 23560 df-hmeo 23753 df-xms 24320 df-ms 24321 df-tms 24322 |
| This theorem is referenced by: stoweidlem29 46017 |
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