Theorem foeq123d 5262

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 5242	. . 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 5243	. . 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 5244	. . 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 213	1 |- ( ph -> ( F : A -onto-> C <-> G : B -onto-> D ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1290   -onto->wfo 5026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-fo 5034

This theorem is referenced by: (None)