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Mirrors > Home > MPE Home > Th. List > coiniss | Structured version Visualization version GIF version |
Description: Coinitiality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.) |
Ref | Expression |
---|---|
cofss.1 | âĒ (ð â ðī â No ) |
cofss.2 | âĒ (ð â ðĩ â ðī) |
Ref | Expression |
---|---|
coiniss | âĒ (ð â âðĨ â ðĩ âðĶ â ðī ðĶ âĪs ðĨ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofss.2 | . . . . 5 âĒ (ð â ðĩ â ðī) | |
2 | 1 | sselda 3977 | . . . 4 âĒ ((ð â§ ð§ â ðĩ) â ð§ â ðī) |
3 | cofss.1 | . . . . . . 7 âĒ (ð â ðī â No ) | |
4 | 1, 3 | sstrd 3987 | . . . . . 6 âĒ (ð â ðĩ â No ) |
5 | 4 | sselda 3977 | . . . . 5 âĒ ((ð â§ ð§ â ðĩ) â ð§ â No ) |
6 | slerflex 27646 | . . . . 5 âĒ (ð§ â No â ð§ âĪs ð§) | |
7 | 5, 6 | syl 17 | . . . 4 âĒ ((ð â§ ð§ â ðĩ) â ð§ âĪs ð§) |
8 | breq1 5144 | . . . . 5 âĒ (ðĶ = ð§ â (ðĶ âĪs ð§ â ð§ âĪs ð§)) | |
9 | 8 | rspcev 3606 | . . . 4 âĒ ((ð§ â ðī â§ ð§ âĪs ð§) â âðĶ â ðī ðĶ âĪs ð§) |
10 | 2, 7, 9 | syl2anc 583 | . . 3 âĒ ((ð â§ ð§ â ðĩ) â âðĶ â ðī ðĶ âĪs ð§) |
11 | 10 | ralrimiva 3140 | . 2 âĒ (ð â âð§ â ðĩ âðĶ â ðī ðĶ âĪs ð§) |
12 | breq2 5145 | . . . 4 âĒ (ðĨ = ð§ â (ðĶ âĪs ðĨ â ðĶ âĪs ð§)) | |
13 | 12 | rexbidv 3172 | . . 3 âĒ (ðĨ = ð§ â (âðĶ â ðī ðĶ âĪs ðĨ â âðĶ â ðī ðĶ âĪs ð§)) |
14 | 13 | cbvralvw 3228 | . 2 âĒ (âðĨ â ðĩ âðĶ â ðī ðĶ âĪs ðĨ â âð§ â ðĩ âðĶ â ðī ðĶ âĪs ð§) |
15 | 11, 14 | sylibr 233 | 1 âĒ (ð â âðĨ â ðĩ âðĶ â ðī ðĶ âĪs ðĨ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 395 â wcel 2098 âwral 3055 âwrex 3064 â wss 3943 class class class wbr 5141 No csur 27523 âĪs csle 27627 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-1o 8464 df-2o 8465 df-no 27526 df-slt 27527 df-sle 27628 |
This theorem is referenced by: cutlt 27802 |
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