Theorem foeq123d 5565

Description: Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
f1eq123d.1	|- ( ph -> F = G )
f1eq123d.2	|- ( ph -> A = B )
f1eq123d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
foeq123d	|- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Proof of Theorem foeq123d

Step	Hyp	Ref	Expression
1		f1eq123d.1	. . 3 |- ( ph -> F = G )
2		foeq1 5544	. . 3 |- ( F = G -> ( F : A -onto-> C <-> G : A -onto-> C ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( F : A -onto-> C <-> G : A -onto-> C ) )
4		f1eq123d.2	. . 3 |- ( ph -> A = B )
5		foeq2 5545	. . 3 |- ( A = B -> ( G : A -onto-> C <-> G : B -onto-> C ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( G : A -onto-> C <-> G : B -onto-> C ) )
7		f1eq123d.3	. . 3 |- ( ph -> C = D )
8		foeq3 5546	. . 3 |- ( C = D -> ( G : B -onto-> C <-> G : B -onto-> D ) )
9	7, 8	syl 14	. 2 |- ( ph -> ( G : B -onto-> C <-> G : B -onto-> D ) )
10	3, 6, 9	3bitrd 214	1 |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   -onto->wfo 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-fo 5324

This theorem is referenced by:  ctssexmid  7317  ctiunctal  13012  unct  13013