Theorem isowe 3888

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 3887	. . 3 |- (H Isom R, S (A, B) -> (R Fr A <-> S Fr B))
2		isorel 3879	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (xRy <-> (H` x)S(H` y)))
3		f1fveq 3861	. . . . . . . 8 |- ((H:A-1-1->B /\ (x e. A /\ y e. A)) -> ((H` x) = (H` y) <-> x = y))
4		isof1o 3878	. . . . . . . . 9 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
5		f1of1 3673	. . . . . . . . 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 448	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> ((H` x) = (H` y) <-> x = y))
8	7	bicomd 519	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (x = y <-> (H` x) = (H` y)))
9		isorel 3879	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (y e. A /\ x e. A)) -> (yRx <-> (H` y)S(H` x)))
10	9	ancom2s 486	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (yRx <-> (H` y)S(H` x)))
11	2, 8, 10	3orbi123d 889	. . . . 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 1670	. . . 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 3680	. . . . 5 |- (H:A-1-1-onto->B -> H:A-onto->B)
14		breq2 2613	. . . . . . . . 9 |- ((H` y) = w -> ((H` x)S(H` y) <-> (H` x)Sw))
15		eqeq2 1476	. . . . . . . . 9 |- ((H` y) = w -> ((H` x) = (H` y) <-> (H` x) = w))
16		breq1 2612	. . . . . . . . 9 |- ((H` y) = w -> ((H` y)S(H` x) <-> wS(H` x)))
17	14, 15, 16	3orbi123d 889	. . . . . . . 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 3870	. . . . . . 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 1655	. . . . . 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 2612	. . . . . . . . 9 |- ((H` x) = z -> ((H` x)Sw <-> zSw))
21		eqeq1 1473	. . . . . . . . 9 |- ((H` x) = z -> ((H` x) = w <-> z = w))
22		breq2 2613	. . . . . . . . 9 |- ((H` x) = z -> (wS(H` x) <-> wSz))
23	20, 21, 22	3orbi123d 889	. . . . . . . 8 |- ((H` x) = z -> (((H` x)Sw \/ (H` x) = w \/ wS(H` x)) <-> (zSw \/ z = w \/ wSz)))
24	23	ralbidv 1655	. . . . . . 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 3870	. . . . . 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 526	. . . . 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 526	. . 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 626	. 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 2925	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
31		dfwe2 2925	. 2 |- (S We B <-> (S Fr B /\ A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
32	29, 30, 31	3bitr4g 553	1 |- (H Isom R, S (A, B) -> (R We A <-> S We B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  A.wral 1637   class class class wbr 2609   Fr wfr 2905   We wwe 2906  -1-1->wf1 3169  -onto->wfo 3170  -1-1-onto->wf1o 3171  ` cfv 3172   Isom wiso 3173

This theorem is referenced by:  f1owe 3890

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-iso 3189

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