Theorem isowe 2941

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 2940	. . 3 ⊢ (H Isom R, S (A, B) → (R Fr A ↔ S Fr B))
2		isorel 2932	. . . . . . . . . 10 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
3		f1fveq 2918	. . . . . . . . . . . 12 ⊢ ((H:A–1-1→B ∧ (x ∈ A ∧ y ∈ A)) → ((H ‘x) = (H ‘y) ↔ x = y))
4		isof1o 2931	. . . . . . . . . . . . 13 ⊢ (H Isom R, S (A, B) → H:A–1-1-onto→B)
5		f1of1 2799	. . . . . . . . . . . . 13 ⊢ (H:A–1-1-onto→B → H:A–1-1→B)
6	4, 5	syl 12	. . . . . . . . . . . 12 ⊢ (H Isom R, S (A, B) → H:A–1-1→B)
7	3, 6	sylan 343	. . . . . . . . . . 11 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ y ∈ A)) → ((H ‘x) = (H ‘y) ↔ x = y))
8	7	bicomd 399	. . . . . . . . . 10 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (x = y ↔ (H ‘x) = (H ‘y)))
9		isorel 2932	. . . . . . . . . . 11 ⊢ ((H Isom R, S (A, B) ∧ (y ∈ A ∧ x ∈ A)) → (yRx ↔ (H ‘y)S(H ‘x)))
10		ancom 333	. . . . . . . . . . 11 ⊢ ((x ∈ A ∧ y ∈ A) ↔ (y ∈ A ∧ x ∈ A))
11	9, 10	sylan2b 347	. . . . . . . . . 10 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (yRx ↔ (H ‘y)S(H ‘x)))
12	2, 8, 11	bi3ord 635	. . . . . . . . 9 ⊢ ((H Isom R, S (A, B) ∧ (x ∈ A ∧ y ∈ A)) → ((xRy ∨ x = y ∨ yRx) ↔ ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))))
13	12	exp32 294	. . . . . . . 8 ⊢ (H Isom R, S (A, B) → (x ∈ A → (y ∈ A → ((xRy ∨ x = y ∨ yRx) ↔ ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))))))
14	13	r19.21adv 1262	. . . . . . 7 ⊢ (H Isom R, S (A, B) → (x ∈ A → ∀y ∈ A ((xRy ∨ x = y ∨ yRx) ↔ ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x)))))
15		r19.15 1292	. . . . . . 7 ⊢ (∀y ∈ A ((xRy ∨ x = y ∨ yRx) ↔ ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))) → (∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))))
16	14, 15	syl6 23	. . . . . 6 ⊢ (H Isom R, S (A, B) → (x ∈ A → (∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x)))))
17	16	r19.21aiv 1259	. . . . 5 ⊢ (H Isom R, S (A, B) → ∀x ∈ A (∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))))
18		r19.15 1292	. . . . 5 ⊢ (∀x ∈ A (∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))) → (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀x ∈ A ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))))
19	17, 18	syl 12	. . . 4 ⊢ (H Isom R, S (A, B) → (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀x ∈ A ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x))))
20		f1ofo 2806	. . . . 5 ⊢ (H:A–1-1-onto→B → H:A–onto→B)
21		breq2 2066	. . . . . . . . 9 ⊢ ((H ‘y) = w → ((H ‘x)S(H ‘y) ↔ (H ‘x)Sw))
22		cleq2 1110	. . . . . . . . 9 ⊢ ((H ‘y) = w → ((H ‘x) = (H ‘y) ↔ (H ‘x) = w))
23		breq1 2065	. . . . . . . . 9 ⊢ ((H ‘y) = w → ((H ‘y)S(H ‘x) ↔ wS(H ‘x)))
24	21, 22, 23	bi3ord 635	. . . . . . . 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))))
25	24	cbvfo 2923	. . . . . . 7 ⊢ (H:A–onto→B → (∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x)) ↔ ∀w ∈ B ((H ‘x)Sw ∨ (H ‘x) = w ∨ wS(H ‘x))))
26	25	biraldv 1219	. . . . . 6 ⊢ (H:A–onto→B → (∀x ∈ A ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x)) ↔ ∀x ∈ A ∀w ∈ B ((H ‘x)Sw ∨ (H ‘x) = w ∨ wS(H ‘x))))
27		breq1 2065	. . . . . . . . 9 ⊢ ((H ‘x) = z → ((H ‘x)Sw ↔ zSw))
28		cleq1 1107	. . . . . . . . 9 ⊢ ((H ‘x) = z → ((H ‘x) = w ↔ z = w))
29		breq2 2066	. . . . . . . . 9 ⊢ ((H ‘x) = z → (wS(H ‘x) ↔ wSz))
30	27, 28, 29	bi3ord 635	. . . . . . . 8 ⊢ ((H ‘x) = z → (((H ‘x)Sw ∨ (H ‘x) = w ∨ wS(H ‘x)) ↔ (zSw ∨ z = w ∨ wSz)))
31	30	biraldv 1219	. . . . . . 7 ⊢ ((H ‘x) = z → (∀w ∈ B ((H ‘x)Sw ∨ (H ‘x) = w ∨ wS(H ‘x)) ↔ ∀w ∈ B (zSw ∨ z = w ∨ wSz)))
32	31	cbvfo 2923	. . . . . 6 ⊢ (H:A–onto→B → (∀x ∈ A ∀w ∈ B ((H ‘x)Sw ∨ (H ‘x) = w ∨ wS(H ‘x)) ↔ ∀z ∈ B ∀w ∈ B (zSw ∨ z = w ∨ wSz)))
33	26, 32	bitrd 406	. . . . 5 ⊢ (H:A–onto→B → (∀x ∈ A ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x)) ↔ ∀z ∈ B ∀w ∈ B (zSw ∨ z = w ∨ wSz)))
34	4, 20, 33	3syl 21	. . . 4 ⊢ (H Isom R, S (A, B) → (∀x ∈ A ∀y ∈ A ((H ‘x)S(H ‘y) ∨ (H ‘x) = (H ‘y) ∨ (H ‘y)S(H ‘x)) ↔ ∀z ∈ B ∀w ∈ B (zSw ∨ z = w ∨ wSz)))
35	19, 34	bitrd 406	. . 3 ⊢ (H Isom R, S (A, B) → (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀z ∈ B ∀w ∈ B (zSw ∨ z = w ∨ wSz)))
36	1, 35	anbi12d 476	. 2 ⊢ (H Isom R, S (A, B) → ((R Fr A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)) ↔ (S Fr B ∧ ∀z ∈ B ∀w ∈ B (zSw ∨ z = w ∨ wSz))))
37		dfwe2 2187	. 2 ⊢ (R We A ↔ (R Fr A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)))
38		dfwe2 2187	. 2 ⊢ (S We B ↔ (S Fr B ∧ ∀z ∈ B ∀w ∈ B (zSw ∨ z = w ∨ wSz)))
39	36, 37, 38	3bitr4g 428	1 ⊢ (H Isom R, S (A, B) → (R We A ↔ S We B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∨ w3o 580   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054   Fr wfr 2061   We wwe 2062  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421   ‘cfv 2422   Isom wiso 2423

This theorem is referenced by:  f1owe 2943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

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