Theorem isowe 3909

Description: An isomorphism preserves well ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33.

Assertion

Ref	Expression
isowe	|- (H Isom R, S (A, B) -> (R We A <-> S We B))

Proof of Theorem isowe

Step	Hyp	Ref	Expression
1		isofr 3908	. . 3 |- (H Isom R, S (A, B) -> (R Fr A <-> S Fr B))
2		isorel 3900	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (xRy <-> (H` x)S(H` y)))
3		f1fveq 3882	. . . . . . . 8 |- ((H:A-1-1->B /\ (x e. A /\ y e. A)) -> ((H` x) = (H` y) <-> x = y))
4		isof1o 3899	. . . . . . . . 9 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
5		f1of1 3694	. . . . . . . . 9 |- (H:A-1-1-onto->B -> H:A-1-1->B)
6	4, 5	syl 10	. . . . . . . 8 |- (H Isom R, S (A, B) -> H:A-1-1->B)
7	3, 6	sylan 450	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> ((H` x) = (H` y) <-> x = y))
8	7	bicomd 523	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (x = y <-> (H` x) = (H` y)))
9		isorel 3900	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (y e. A /\ x e. A)) -> (yRx <-> (H` y)S(H` x)))
10	9	ancom2s 489	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (yRx <-> (H` y)S(H` x)))
11	2, 8, 10	3orbi123d 894	. . . . 5 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> ((xRy \/ x = y \/ yRx) <-> ((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x))))
12	11	2ralbidva 1681	. . . 4 |- (H Isom R, S (A, B) -> (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.x e. A A.y e. A ((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x))))
13		f1ofo 3701	. . . . 5 |- (H:A-1-1-onto->B -> H:A-onto->B)
14		breq2 2628	. . . . . . . . 9 |- ((H` y) = w -> ((H` x)S(H` y) <-> (H` x)Sw))
15		eqeq2 1487	. . . . . . . . 9 |- ((H` y) = w -> ((H` x) = (H` y) <-> (H` x) = w))
16		breq1 2627	. . . . . . . . 9 |- ((H` y) = w -> ((H` y)S(H` x) <-> wS(H` x)))
17	14, 15, 16	3orbi123d 894	. . . . . . . 8 |- ((H` y) = w -> (((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x)) <-> ((H` x)Sw \/ (H` x) = w \/ wS(H` x))))
18	17	cbvfo 3891	. . . . . . 7 |- (H:A-onto->B -> (A.y e. A ((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x)) <-> A.w e. B ((H` x)Sw \/ (H` x) = w \/ wS(H` x))))
19	18	ralbidv 1666	. . . . . 6 |- (H:A-onto->B -> (A.x e. A A.y e. A ((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x)) <-> A.x e. A A.w e. B ((H` x)Sw \/ (H` x) = w \/ wS(H` x))))
20		breq1 2627	. . . . . . . . 9 |- ((H` x) = z -> ((H` x)Sw <-> zSw))
21		eqeq1 1484	. . . . . . . . 9 |- ((H` x) = z -> ((H` x) = w <-> z = w))
22		breq2 2628	. . . . . . . . 9 |- ((H` x) = z -> (wS(H` x) <-> wSz))
23	20, 21, 22	3orbi123d 894	. . . . . . . 8 |- ((H` x) = z -> (((H` x)Sw \/ (H` x) = w \/ wS(H` x)) <-> (zSw \/ z = w \/ wSz)))
24	23	ralbidv 1666	. . . . . . 7 |- ((H` x) = z -> (A.w e. B ((H` x)Sw \/ (H` x) = w \/ wS(H` x)) <-> A.w e. B (zSw \/ z = w \/ wSz)))
25	24	cbvfo 3891	. . . . . 6 |- (H:A-onto->B -> (A.x e. A A.w e. B ((H` x)Sw \/ (H` x) = w \/ wS(H` x)) <-> A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
26	19, 25	bitrd 530	. . . . 5 |- (H:A-onto->B -> (A.x e. A A.y e. A ((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x)) <-> A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
27	4, 13, 26	3syl 20	. . . 4 |- (H Isom R, S (A, B) -> (A.x e. A A.y e. A ((H` x)S(H` y) \/ (H` x) = (H` y) \/ (H` y)S(H` x)) <-> A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
28	12, 27	bitrd 530	. . 3 |- (H Isom R, S (A, B) -> (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
29	1, 28	anbi12d 630	. 2 |- (H Isom R, S (A, B) -> ((R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)) <-> (S Fr B /\ A.z e. B A.w e. B (zSw \/ z = w \/ wSz))))
30		dfwe2 2941	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
31		dfwe2 2941	. 2 |- (S We B <-> (S Fr B /\ A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
32	29, 30, 31	3bitr4g 557	1 |- (H Isom R, S (A, B) -> (R We A <-> S We B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960  A.wral 1648   class class class wbr 2624   Fr wfr 2921   We wwe 2922  -1-1->wf1 3185  -onto->wfo 3186  -1-1-onto->wf1o 3187  ` cfv 3188   Isom wiso 3189

This theorem is referenced by:  f1owe 3911

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-iso 3205

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