Theorem mapss 6940

Description: Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
mapss	|- ( ( B e. V /\ A C_ B ) -> ( A ^m C ) C_ ( B ^m C ) )

Proof of Theorem mapss

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 6918	. . . . . 6 |- ( f e. ( A ^m C ) -> f : C --> A )
2	1	adantl 277	. . . . 5 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> f : C --> A )
3		simplr 529	. . . . 5 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> A C_ B )
4	2, 3	fssd 5528	. . . 4 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> f : C --> B )
5		simpll 527	. . . . 5 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> B e. V )
6		elmapex 6917	. . . . . . 7 |- ( f e. ( A ^m C ) -> ( A e. _V /\ C e. _V ) )
7	6	simprd 114	. . . . . 6 |- ( f e. ( A ^m C ) -> C e. _V )
8	7	adantl 277	. . . . 5 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> C e. _V )
9	5, 8	elmapd 6910	. . . 4 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> ( f e. ( B ^m C ) <-> f : C --> B ) )
10	4, 9	mpbird 167	. . 3 |- ( ( ( B e. V /\ A C_ B ) /\ f e. ( A ^m C ) ) -> f e. ( B ^m C ) )
11	10	ex 115	. 2 |- ( ( B e. V /\ A C_ B ) -> ( f e. ( A ^m C ) -> f e. ( B ^m C ) ) )
12	11	ssrdv 3248	1 |- ( ( B e. V /\ A C_ B ) -> ( A ^m C ) C_ ( B ^m C ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   _Vcvv 2815   C_ wss 3214   -->wf 5354  (class class class)co 6059   ^m cmap 6896

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-map 6898

This theorem is referenced by:  mapdom1g  7114  plyss  15734  bj-charfunbi  16722