Theorem k0004lem1 38445

Description: Application of ssin 3835 to range of a function. (Contributed by RP, 1-Apr-2021.)

Assertion

Ref	Expression
k0004lem1	|- ( D = ( B i^i C ) -> ( ( F : A --> B /\ ( F " A ) C_ C ) <-> F : A --> D ) )

Proof of Theorem k0004lem1

Step	Hyp	Ref	Expression
1		feq3 6028	. 2 |- ( D = ( B i^i C ) -> ( F : A --> D <-> F : A --> ( B i^i C ) ) )
2		fnima 6010	. . . . . . 7 |- ( F Fn A -> ( F " A ) = ran F )
3	2	sseq1d 3632	. . . . . 6 |- ( F Fn A -> ( ( F " A ) C_ C <-> ran F C_ C ) )
4	3	anbi2d 740	. . . . 5 |- ( F Fn A -> ( ( ran F C_ B /\ ( F " A ) C_ C ) <-> ( ran F C_ B /\ ran F C_ C ) ) )
5		ssin 3835	. . . . 5 |- ( ( ran F C_ B /\ ran F C_ C ) <-> ran F C_ ( B i^i C ) )
6	4, 5	syl6bb 276	. . . 4 |- ( F Fn A -> ( ( ran F C_ B /\ ( F " A ) C_ C ) <-> ran F C_ ( B i^i C ) ) )
7	6	pm5.32i 669	. . 3 |- ( ( F Fn A /\ ( ran F C_ B /\ ( F " A ) C_ C ) ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
8		df-f 5892	. . . . 5 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	8	anbi1i 731	. . . 4 |- ( ( F : A --> B /\ ( F " A ) C_ C ) <-> ( ( F Fn A /\ ran F C_ B ) /\ ( F " A ) C_ C ) )
10		anass 681	. . . 4 |- ( ( ( F Fn A /\ ran F C_ B ) /\ ( F " A ) C_ C ) <-> ( F Fn A /\ ( ran F C_ B /\ ( F " A ) C_ C ) ) )
11	9, 10	bitri 264	. . 3 |- ( ( F : A --> B /\ ( F " A ) C_ C ) <-> ( F Fn A /\ ( ran F C_ B /\ ( F " A ) C_ C ) ) )
12		df-f 5892	. . 3 |- ( F : A --> ( B i^i C ) <-> ( F Fn A /\ ran F C_ ( B i^i C ) ) )
13	7, 11, 12	3bitr4i 292	. 2 |- ( ( F : A --> B /\ ( F " A ) C_ C ) <-> F : A --> ( B i^i C ) )
14	1, 13	syl6rbbr 279	1 |- ( D = ( B i^i C ) -> ( ( F : A --> B /\ ( F " A ) C_ C ) <-> F : A --> D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   i^i cin 3573   C_ wss 3574   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  k0004lem2  38446