Theorem uniimadomf 9367

Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9366 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)

Hypotheses

Ref	Expression
uniimadomf.1	|- F/_ x F
uniimadomf.2	|- A e. _V
uniimadomf.3	|- B e. _V

Assertion

Ref	Expression
uniimadomf	|- ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> U. ( F " A ) ~<_ ( A X. B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   F( x)

Proof of Theorem uniimadomf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ z ( F ` x ) ~<_ B
2		uniimadomf.1	. . . . 5 |- F/_ x F
3		nfcv 2764	. . . . 5 |- F/_ x z
4	2, 3	nffv 6198	. . . 4 |- F/_ x ( F ` z )
5		nfcv 2764	. . . 4 |- F/_ x ~<_
6		nfcv 2764	. . . 4 |- F/_ x B
7	4, 5, 6	nfbr 4699	. . 3 |- F/ x ( F ` z ) ~<_ B
8		fveq2 6191	. . . 4 |- ( x = z -> ( F ` x ) = ( F ` z ) )
9	8	breq1d 4663	. . 3 |- ( x = z -> ( ( F ` x ) ~<_ B <-> ( F ` z ) ~<_ B ) )
10	1, 7, 9	cbvral 3167	. 2 |- ( A. x e. A ( F ` x ) ~<_ B <-> A. z e. A ( F ` z ) ~<_ B )
11		uniimadomf.2	. . 3 |- A e. _V
12		uniimadomf.3	. . 3 |- B e. _V
13	11, 12	uniimadom 9366	. 2 |- ( ( Fun F /\ A. z e. A ( F ` z ) ~<_ B ) -> U. ( F " A ) ~<_ ( A X. B ) )
14	10, 13	sylan2b 492	1 |- ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> U. ( F " A ) ~<_ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   U.cuni 4436   class class class wbr 4653   X. cxp 5112   "cima 5117   Fun wfun 5882   ` cfv 5888   ~<_ cdom 7953

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by: (None)