Theorem iundom 5974

Description: An upper bound for the cardinality of an indexed union. C depends on x and should be thought of as C(x).

Hypotheses

Ref	Expression
iundom.1	|- A e. _V
iundom.2	|- B e. _V
iundom.3	|- C e. _V

Assertion

Ref	Expression
iundom	|- (A.x e. A C ~<_ B -> U_x e. A C ~<_ (A X. B))

Distinct variable groups:   x,A   x,B

Proof of Theorem iundom

Step	Hyp	Ref	Expression
1		iundom.3	. . . . 5 |- C e. _V
2		fvopab2 4754	. . . . 5 |- ((x e. A /\ C e. _V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
3	1, 2	mpan2 760	. . . 4 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
4	3	breq1d 3348	. . 3 |- (x e. A -> (({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B <-> C ~<_ B))
5	4	ralbiia 2133	. 2 |- (A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B <-> A.x e. A C ~<_ B)
6		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (x e. A /\ y = C)}
7	1, 6	fnopab2 4549	. . . . 5 |- {<.x, y>. | (x e. A /\ y = C)} Fn A
8		fnfun 4510	. . . . 5 |- ({<.x, y>. | (x e. A /\ y = C)} Fn A -> Fun {<.x, y>. | (x e. A /\ y = C)})
9	7, 8	ax-mp 7	. . . 4 |- Fun {<.x, y>. | (x e. A /\ y = C)}
10		hbopab1 3562	. . . . 5 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
11		iundom.1	. . . . 5 |- A e. _V
12		iundom.2	. . . . 5 |- B e. _V
13	10, 11, 12	uniimadomf 5973	. . . 4 |- ((Fun {<.x, y>. | (x e. A /\ y = C)} /\ A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B) -> U.({<.x, y>. | (x e. A /\ y = C)}"A) ~<_ (A X. B))
14	9, 13	mpan 759	. . 3 |- (A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B -> U.({<.x, y>. | (x e. A /\ y = C)}"A) ~<_ (A X. B))
15	3	iuneq2i 3276	. . . 4 |- U_x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) = U_x e. A C
16	10	funiunfvf 4846	. . . . 5 |- (Fun {<.x, y>. | (x e. A /\ y = C)} -> U_x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) = U.({<.x, y>. | (x e. A /\ y = C)}"A))
17	9, 16	ax-mp 7	. . . 4 |- U_x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) = U.({<.x, y>. | (x e. A /\ y = C)}"A)
18	15, 17	eqtr3i 1910	. . 3 |- U_x e. A C = U.({<.x, y>. | (x e. A /\ y = C)}"A)
19	14, 18	syl5eqbr 3370	. 2 |- (A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B -> U_x e. A C ~<_ (A X. B))
20	5, 19	sylbir 218	1 |- (A.x e. A C ~<_ B -> U_x e. A C ~<_ (A X. B))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  U.cuni 3177  U_ciun 3255   class class class wbr 3338  {copab 3395   X. cxp 3984  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998   ~<_ cdom 5424

This theorem is referenced by:  iunctb 8844

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-en 5427  df-dom 5428  df-r1 5750  df-rank 5751

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