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Mirrors > Home > MPE Home > Th. List > hsmexlem3 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10429. Clear πΌ hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g., using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
hsmexlem.f | β’ πΉ = OrdIso( E , π΅) |
hsmexlem.g | β’ πΊ = OrdIso( E , βͺ π β π΄ π΅) |
Ref | Expression |
---|---|
hsmexlem3 | β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β dom πΊ β (harβπ« (π· Γ πΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wdomref 9569 | . . . . 5 β’ (πΆ β On β πΆ βΌ* πΆ) | |
2 | xpwdomg 9582 | . . . . 5 β’ ((π΄ βΌ* π· β§ πΆ βΌ* πΆ) β (π΄ Γ πΆ) βΌ* (π· Γ πΆ)) | |
3 | 1, 2 | sylan2 591 | . . . 4 β’ ((π΄ βΌ* π· β§ πΆ β On) β (π΄ Γ πΆ) βΌ* (π· Γ πΆ)) |
4 | wdompwdom 9575 | . . . 4 β’ ((π΄ Γ πΆ) βΌ* (π· Γ πΆ) β π« (π΄ Γ πΆ) βΌ π« (π· Γ πΆ)) | |
5 | harword 9560 | . . . 4 β’ (π« (π΄ Γ πΆ) βΌ π« (π· Γ πΆ) β (harβπ« (π΄ Γ πΆ)) β (harβπ« (π· Γ πΆ))) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 β’ ((π΄ βΌ* π· β§ πΆ β On) β (harβπ« (π΄ Γ πΆ)) β (harβπ« (π· Γ πΆ))) |
7 | 6 | adantr 479 | . 2 β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β (harβπ« (π΄ Γ πΆ)) β (harβπ« (π· Γ πΆ))) |
8 | relwdom 9563 | . . . . . 6 β’ Rel βΌ* | |
9 | 8 | brrelex1i 5731 | . . . . 5 β’ (π΄ βΌ* π· β π΄ β V) |
10 | 9 | adantr 479 | . . . 4 β’ ((π΄ βΌ* π· β§ πΆ β On) β π΄ β V) |
11 | 10 | adantr 479 | . . 3 β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β π΄ β V) |
12 | simplr 765 | . . 3 β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β πΆ β On) | |
13 | simpr 483 | . . 3 β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) | |
14 | hsmexlem.f | . . . 4 β’ πΉ = OrdIso( E , π΅) | |
15 | hsmexlem.g | . . . 4 β’ πΊ = OrdIso( E , βͺ π β π΄ π΅) | |
16 | 14, 15 | hsmexlem2 10424 | . . 3 β’ ((π΄ β V β§ πΆ β On β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β dom πΊ β (harβπ« (π΄ Γ πΆ))) |
17 | 11, 12, 13, 16 | syl3anc 1369 | . 2 β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β dom πΊ β (harβπ« (π΄ Γ πΆ))) |
18 | 7, 17 | sseldd 3982 | 1 β’ (((π΄ βΌ* π· β§ πΆ β On) β§ βπ β π΄ (π΅ β π« On β§ dom πΉ β πΆ)) β dom πΊ β (harβπ« (π· Γ πΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 βwral 3059 Vcvv 3472 β wss 3947 π« cpw 4601 βͺ ciun 4996 class class class wbr 5147 E cep 5578 Γ cxp 5673 dom cdm 5675 Oncon0 6363 βcfv 6542 βΌ cdom 8939 OrdIsocoi 9506 harchar 9553 βΌ* cwdom 9561 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-smo 8348 df-recs 8373 df-en 8942 df-dom 8943 df-sdom 8944 df-oi 9507 df-har 9554 df-wdom 9562 |
This theorem is referenced by: hsmexlem4 10426 hsmexlem5 10427 |
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