Theorem surrc2 14390

Description: A surjection is right-cancelable.

Assertion

Ref	Expression
surrc2	|- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> ((H o. F) = (K o. F) <-> H = K))

Proof of Theorem surrc2

Step	Hyp	Ref	Expression
1		eqtr3 1907	. . . . . . . 8 |- ((dom H = B /\ dom K = B) -> dom H = dom K)
2		fndm 4512	. . . . . . . 8 |- (H Fn B -> dom H = B)
3		fndm 4512	. . . . . . . 8 |- (K Fn B -> dom K = B)
4	1, 2, 3	syl2an 503	. . . . . . 7 |- ((H Fn B /\ K Fn B) -> dom H = dom K)
5	4	3adant1 894	. . . . . 6 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> dom H = dom K)
6	5	adantr 425	. . . . 5 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> dom H = dom K)
7		fveq1 4680	. . . . . . . . 9 |- ((H o. F) = (K o. F) -> ((H o. F)` y) = ((K o. F)` y))
8		fvco3 4739	. . . . . . . . . . . . . . . . . 18 |- ((Fun H /\ F:A-->B /\ y e. A) -> ((H o. F)` y) = (H` (F` y)))
9	8	3exp 1066	. . . . . . . . . . . . . . . . 17 |- (Fun H -> (F:A-->B -> (y e. A -> ((H o. F)` y) = (H` (F` y)))))
10	9	impcom 378	. . . . . . . . . . . . . . . 16 |- ((F:A-->B /\ Fun H) -> (y e. A -> ((H o. F)` y) = (H` (F` y))))
11		fof 4617	. . . . . . . . . . . . . . . 16 |- (F:A-onto->B -> F:A-->B)
12		fnfun 4510	. . . . . . . . . . . . . . . 16 |- (H Fn B -> Fun H)
13	10, 11, 12	syl2an 503	. . . . . . . . . . . . . . 15 |- ((F:A-onto->B /\ H Fn B) -> (y e. A -> ((H o. F)` y) = (H` (F` y))))
14	13	3adant3 896	. . . . . . . . . . . . . 14 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> (y e. A -> ((H o. F)` y) = (H` (F` y))))
15	14	imp 377	. . . . . . . . . . . . 13 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ y e. A) -> ((H o. F)` y) = (H` (F` y)))
16		fvco3 4739	. . . . . . . . . . . . . . . . . . 19 |- ((Fun K /\ F:A-->B /\ y e. A) -> ((K o. F)` y) = (K` (F` y)))
17	16	3exp 1066	. . . . . . . . . . . . . . . . . 18 |- (Fun K -> (F:A-->B -> (y e. A -> ((K o. F)` y) = (K` (F` y)))))
18	17, 11	syl5com 63	. . . . . . . . . . . . . . . . 17 |- (F:A-onto->B -> (Fun K -> (y e. A -> ((K o. F)` y) = (K` (F` y)))))
19		fnfun 4510	. . . . . . . . . . . . . . . . 17 |- (K Fn B -> Fun K)
20	18, 19	syl5 20	. . . . . . . . . . . . . . . 16 |- (F:A-onto->B -> (K Fn B -> (y e. A -> ((K o. F)` y) = (K` (F` y)))))
21	20	imp 377	. . . . . . . . . . . . . . 15 |- ((F:A-onto->B /\ K Fn B) -> (y e. A -> ((K o. F)` y) = (K` (F` y))))
22	21	3adant2 895	. . . . . . . . . . . . . 14 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> (y e. A -> ((K o. F)` y) = (K` (F` y))))
23	22	imp 377	. . . . . . . . . . . . 13 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ y e. A) -> ((K o. F)` y) = (K` (F` y)))
24	15, 23	eqeq12d 1899	. . . . . . . . . . . 12 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ y e. A) -> (((H o. F)` y) = ((K o. F)` y) <-> (H` (F` y)) = (K` (F` y))))
25	24	biimpd 170	. . . . . . . . . . 11 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ y e. A) -> (((H o. F)` y) = ((K o. F)` y) -> (H` (F` y)) = (K` (F` y))))
26	25	ex 402	. . . . . . . . . 10 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> (y e. A -> (((H o. F)` y) = ((K o. F)` y) -> (H` (F` y)) = (K` (F` y)))))
27	26	com3r 39	. . . . . . . . 9 |- (((H o. F)` y) = ((K o. F)` y) -> ((F:A-onto->B /\ H Fn B /\ K Fn B) -> (y e. A -> (H` (F` y)) = (K` (F` y)))))
28	7, 27	syl 12	. . . . . . . 8 |- ((H o. F) = (K o. F) -> ((F:A-onto->B /\ H Fn B /\ K Fn B) -> (y e. A -> (H` (F` y)) = (K` (F` y)))))
29	28	impcom 378	. . . . . . 7 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> (y e. A -> (H` (F` y)) = (K` (F` y))))
30	29	r19.21aiv 2175	. . . . . 6 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> A.y e. A (H` (F` y)) = (K` (F` y)))
31		foeq3 4615	. . . . . . . . . . . . 13 |- (B = dom H -> (F:A-onto->B <-> F:A-onto->dom H))
32	31	eqcoms 1887	. . . . . . . . . . . 12 |- (dom H = B -> (F:A-onto->B <-> F:A-onto->dom H))
33	32	biimpd 170	. . . . . . . . . . 11 |- (dom H = B -> (F:A-onto->B -> F:A-onto->dom H))
34	2, 33	syl 12	. . . . . . . . . 10 |- (H Fn B -> (F:A-onto->B -> F:A-onto->dom H))
35	34	impcom 378	. . . . . . . . 9 |- ((F:A-onto->B /\ H Fn B) -> F:A-onto->dom H)
36	35	3adant3 896	. . . . . . . 8 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> F:A-onto->dom H)
37	36	adantr 425	. . . . . . 7 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> F:A-onto->dom H)
38		fveq2 4681	. . . . . . . . 9 |- ((F` y) = x -> (H` (F` y)) = (H` x))
39		fveq2 4681	. . . . . . . . 9 |- ((F` y) = x -> (K` (F` y)) = (K` x))
40	38, 39	eqeq12d 1899	. . . . . . . 8 |- ((F` y) = x -> ((H` (F` y)) = (K` (F` y)) <-> (H` x) = (K` x)))
41	40	cbvfo 4861	. . . . . . 7 |- (F:A-onto->dom H -> (A.y e. A (H` (F` y)) = (K` (F` y)) <-> A.x e. dom H(H` x) = (K` x)))
42	37, 41	syl 12	. . . . . 6 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> (A.y e. A (H` (F` y)) = (K` (F` y)) <-> A.x e. dom H(H` x) = (K` x)))
43	30, 42	mpbid 212	. . . . 5 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> A.x e. dom H(H` x) = (K` x))
44	6, 43	jca 310	. . . 4 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> (dom H = dom K /\ A.x e. dom H(H` x) = (K` x)))
45	12, 19	anim12i 360	. . . . . . 7 |- ((H Fn B /\ K Fn B) -> (Fun H /\ Fun K))
46	45	3adant1 894	. . . . . 6 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> (Fun H /\ Fun K))
47	46	adantr 425	. . . . 5 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> (Fun H /\ Fun K))
48		eqfunfv 13839	. . . . 5 |- ((Fun H /\ Fun K) -> (H = K <-> (dom H = dom K /\ A.x e. dom H(H` x) = (K` x))))
49	47, 48	syl 12	. . . 4 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> (H = K <-> (dom H = dom K /\ A.x e. dom H(H` x) = (K` x))))
50	44, 49	mpbird 213	. . 3 |- (((F:A-onto->B /\ H Fn B /\ K Fn B) /\ (H o. F) = (K o. F)) -> H = K)
51	50	ex 402	. 2 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> ((H o. F) = (K o. F) -> H = K))
52		coeq1 4123	. 2 |- (H = K -> (H o. F) = (K o. F))
53	51, 52	impbid1 575	1 |- ((F:A-onto->B /\ H Fn B /\ K Fn B) -> ((H o. F) = (K o. F) <-> H = K))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  dom cdm 3986   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998

This theorem is referenced by:  injsurinj 14487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014

Copyright terms: Public domain