Theorem f1oresrab 5272

Description: Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)

Hypotheses

Ref	Expression
f1oresrab.1	|- F = (x e. A |-> C)
f1oresrab.2	|- (ph -> F :A -1-1-onto->B)
f1oresrab.3	|- ((ph /\ x e. A /\ y = C) -> (ch <-> ps))

Assertion

Ref	Expression
f1oresrab	|- (ph -> ( F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch })

Distinct variable groups:   x,y,A   x,B,y   y, C   ph,x,y   ps,y   ch,x

Allowed substitution hints:   ps(x)   ch(y)   C(x)   F(x,y)

Proof of Theorem f1oresrab

Step	Hyp	Ref	Expression
1		f1oresrab.2	. . . 4 |- (ph -> F :A -1-1-onto->B)
2		f1ofun 5071	. . . 4 |- ( F :A -1-1-onto->B -> Fun F)
3		funcnvcnv 4901	. . . 4 |- ( Fun F -> Fun `' `' F)
4	1, 2, 3	3syl 17	. . 3 |- (ph -> Fun `' `' F)
5		f1ocnv 5082	. . . . . . 7 |- ( F :A -1-1-onto->B -> `' F :B -1-1-onto->A)
6	1, 5	syl 14	. . . . . 6 |- (ph -> `' F :B -1-1-onto->A)
7		f1of1 5068	. . . . . 6 |- ( `' F :B -1-1-onto->A -> `' F :B -1-1->A)
8	6, 7	syl 14	. . . . 5 |- (ph -> `' F :B -1-1->A)
9		ssrab2 3019	. . . . 5 |- {y e. B | ch } C_ B
10		f1ores 5084	. . . . 5 |- (( `' F :B -1-1->A /\ {y e. B | ch } C_ B) -> ( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto->( `' F " {y e. B | ch }))
11	8, 9, 10	sylancl 392	. . . 4 |- (ph -> ( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto->( `' F " {y e. B | ch }))
12		f1oresrab.1	. . . . . . 7 |- F = (x e. A |-> C)
13	12	mptpreima 4757	. . . . . 6 |- ( `' F " {y e. B | ch }) = {x e. A | C e. {y e. B | ch } }
14		f1oresrab.3	. . . . . . . . . 10 |- ((ph /\ x e. A /\ y = C) -> (ch <-> ps))
15	14	3expia 1105	. . . . . . . . 9 |- ((ph /\ x e. A) -> (y = C -> (ch <-> ps)))
16	15	alrimiv 1751	. . . . . . . 8 |- ((ph /\ x e. A) -> A.y(y = C -> (ch <-> ps)))
17		f1of 5069	. . . . . . . . . . 11 |- ( F :A -1-1-onto->B -> F :A -->B)
18	1, 17	syl 14	. . . . . . . . . 10 |- (ph -> F :A -->B)
19	12	fmpt 5262	. . . . . . . . . 10 |- (A.x e. A C e. B <-> F :A -->B)
20	18, 19	sylibr 137	. . . . . . . . 9 |- (ph -> A.x e. A C e. B)
21	20	r19.21bi 2401	. . . . . . . 8 |- ((ph /\ x e. A) -> C e. B)
22		elrab3t 2691	. . . . . . . 8 |- ((A.y(y = C -> (ch <-> ps)) /\ C e. B) -> ( C e. {y e. B | ch } <-> ps))
23	16, 21, 22	syl2anc 391	. . . . . . 7 |- ((ph /\ x e. A) -> ( C e. {y e. B | ch } <-> ps))
24	23	rabbidva 2542	. . . . . 6 |- (ph -> {x e. A | C e. {y e. B | ch } } = {x e. A | ps })
25	13, 24	syl5eq 2081	. . . . 5 |- (ph -> ( `' F " {y e. B | ch }) = {x e. A | ps })
26		f1oeq3 5062	. . . . 5 |- (( `' F " {y e. B | ch }) = {x e. A | ps } -> (( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto->( `' F " {y e. B | ch }) <-> ( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto-> {x e. A | ps }))
27	25, 26	syl 14	. . . 4 |- (ph -> (( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto->( `' F " {y e. B | ch }) <-> ( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto-> {x e. A | ps }))
28	11, 27	mpbid 135	. . 3 |- (ph -> ( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto-> {x e. A | ps })
29		f1orescnv 5085	. . 3 |- (( Fun `' `' F /\ ( `' F |` {y e. B | ch }) : {y e. B | ch } -1-1-onto-> {x e. A | ps }) -> ( `' `' F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch })
30	4, 28, 29	syl2anc 391	. 2 |- (ph -> ( `' `' F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch })
31		rescnvcnv 4726	. . 3 |- ( `' `' F |` {x e. A | ps }) = ( F |` {x e. A | ps })
32		f1oeq1 5060	. . 3 |- (( `' `' F |` {x e. A | ps }) = ( F |` {x e. A | ps }) -> (( `' `' F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch } <-> ( F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch }))
33	31, 32	ax-mp 7	. 2 |- (( `' `' F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch } <-> ( F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch })
34	30, 33	sylib 127	1 |- (ph -> ( F |` {x e. A | ps }) : {x e. A | ps } -1-1-onto-> {y e. B | ch })

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884  A.wal 1240   = wceq 1242   e. wcel 1390  A.wral 2300   {crab 2304   C_ wss 2911   |-> cmpt 3809   `'ccnv 4287   |` cres 4290   "cima 4291   Fun wfun 4839   -->wf 4841   -1-1->wf1 4842   -1-1-onto->wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by: (None)