Theorem isowe 3903

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 3902	. . 3 |- (H Isom R, S (A, B) -> (R Fr A <-> S Fr B))
2		isorel 3894	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (xRy <-> (H` x)S(H` y)))
3		f1fveq 3876	. . . . . . . 8 |- ((H:A-1-1->B /\ (x e. A /\ y e. A)) -> ((H` x) = (H` y) <-> x = y))
4		isof1o 3893	. . . . . . . . 9 |- (H Isom R, S (A, B) -> H:A-1-1-onto->B)
5		f1of1 3688	. . . . . . . . 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 521	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (x = y <-> (H` x) = (H` y)))
9		isorel 3894	. . . . . . 7 |- ((H Isom R, S (A, B) /\ (y e. A /\ x e. A)) -> (yRx <-> (H` y)S(H` x)))
10	9	ancom2s 487	. . . . . 6 |- ((H Isom R, S (A, B) /\ (x e. A /\ y e. A)) -> (yRx <-> (H` y)S(H` x)))
11	2, 8, 10	3orbi123d 892	. . . . 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 1678	. . . 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 3695	. . . . 5 |- (H:A-1-1-onto->B -> H:A-onto->B)
14		breq2 2623	. . . . . . . . 9 |- ((H` y) = w -> ((H` x)S(H` y) <-> (H` x)Sw))
15		eqeq2 1484	. . . . . . . . 9 |- ((H` y) = w -> ((H` x) = (H` y) <-> (H` x) = w))
16		breq1 2622	. . . . . . . . 9 |- ((H` y) = w -> ((H` y)S(H` x) <-> wS(H` x)))
17	14, 15, 16	3orbi123d 892	. . . . . . . 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 3885	. . . . . . 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 1663	. . . . . 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 2622	. . . . . . . . 9 |- ((H` x) = z -> ((H` x)Sw <-> zSw))
21		eqeq1 1481	. . . . . . . . 9 |- ((H` x) = z -> ((H` x) = w <-> z = w))
22		breq2 2623	. . . . . . . . 9 |- ((H` x) = z -> (wS(H` x) <-> wSz))
23	20, 21, 22	3orbi123d 892	. . . . . . . 8 |- ((H` x) = z -> (((H` x)Sw \/ (H` x) = w \/ wS(H` x)) <-> (zSw \/ z = w \/ wSz)))
24	23	ralbidv 1663	. . . . . . 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 3885	. . . . . 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 528	. . . . 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 528	. . 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 628	. 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 2935	. 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
31		dfwe2 2935	. 2 |- (S We B <-> (S Fr B /\ A.z e. B A.w e. B (zSw \/ z = w \/ wSz)))
32	29, 30, 31	3bitr4g 555	1 |- (H Isom R, S (A, B) -> (R We A <-> S We B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 774   = wceq 956   e. wcel 958  A.wral 1645   class class class wbr 2619   Fr wfr 2915   We wwe 2916  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181  ` cfv 3182   Isom wiso 3183

This theorem is referenced by:  f1owe 3905

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-iso 3199

Copyright terms: Public domain