Theorem abrexdom 15739

Description: An indexed set is dominated by the indexing set.

Hypothesis

Ref	Expression
abrexdom.1	|- (y e. A -> E*xph)

Assertion

Ref	Expression
abrexdom	|- (A e. C -> {x | E.y e. A ph} ~<_ A)

Distinct variable group:   x,A,y

Proof of Theorem abrexdom

Step	Hyp	Ref	Expression
1		dmopabss 4168	. . . . 5 |- dom {<.y, x>. | (y e. A /\ ph)} C_ A
2		ssexg 3457	. . . . 5 |- ((dom {<.y, x>. | (y e. A /\ ph)} C_ A /\ A e. C) -> dom {<.y, x>. | (y e. A /\ ph)} e. _V)
3	1, 2	mpan 759	. . . 4 |- (A e. C -> dom {<.y, x>. | (y e. A /\ ph)} e. _V)
4		funopab 4455	. . . . . . 7 |- (Fun {<.y, x>. | (y e. A /\ ph)} <-> A.yE*x(y e. A /\ ph))
5		abrexdom.1	. . . . . . . 8 |- (y e. A -> E*xph)
6		moanimv 1829	. . . . . . . 8 |- (E*x(y e. A /\ ph) <-> (y e. A -> E*xph))
7	5, 6	mpbir 207	. . . . . . 7 |- E*x(y e. A /\ ph)
8	4, 7	mpgbir 1334	. . . . . 6 |- Fun {<.y, x>. | (y e. A /\ ph)}
9	8	a1i 8	. . . . 5 |- (A e. C -> Fun {<.y, x>. | (y e. A /\ ph)})
10		funfn 4451	. . . . 5 |- (Fun {<.y, x>. | (y e. A /\ ph)} <-> {<.y, x>. | (y e. A /\ ph)} Fn dom {<.y, x>. | (y e. A /\ ph)})
11	9, 10	sylib 215	. . . 4 |- (A e. C -> {<.y, x>. | (y e. A /\ ph)} Fn dom {<.y, x>. | (y e. A /\ ph)})
12		fnrndomg 5969	. . . 4 |- (dom {<.y, x>. | (y e. A /\ ph)} e. _V -> ({<.y, x>. | (y e. A /\ ph)} Fn dom {<.y, x>. | (y e. A /\ ph)} -> ran {<.y, x>. | (y e. A /\ ph)} ~<_ dom {<.y, x>. | (y e. A /\ ph)}))
13	3, 11, 12	sylc 83	. . 3 |- (A e. C -> ran {<.y, x>. | (y e. A /\ ph)} ~<_ dom {<.y, x>. | (y e. A /\ ph)})
14		ssdom2g 5468	. . . 4 |- (A e. C -> (dom {<.y, x>. | (y e. A /\ ph)} C_ A -> dom {<.y, x>. | (y e. A /\ ph)} ~<_ A))
15	1, 14	mpi 55	. . 3 |- (A e. C -> dom {<.y, x>. | (y e. A /\ ph)} ~<_ A)
16		domtr 5474	. . 3 |- ((ran {<.y, x>. | (y e. A /\ ph)} ~<_ dom {<.y, x>. | (y e. A /\ ph)} /\ dom {<.y, x>. | (y e. A /\ ph)} ~<_ A) -> ran {<.y, x>. | (y e. A /\ ph)} ~<_ A)
17	13, 15, 16	syl11anc 524	. 2 |- (A e. C -> ran {<.y, x>. | (y e. A /\ ph)} ~<_ A)
18		df-rex 2110	. . . 4 |- (E.y e. A ph <-> E.y(y e. A /\ ph))
19	18	abbii 2006	. . 3 |- {x | E.y e. A ph} = {x | E.y(y e. A /\ ph)}
20		rnopab 4201	. . 3 |- ran {<.y, x>. | (y e. A /\ ph)} = {x | E.y(y e. A /\ ph)}
21	19, 20	eqtr4i 1911	. 2 |- {x | E.y e. A ph} = ran {<.y, x>. | (y e. A /\ ph)}
22	17, 21	syl5eqbr 3370	1 |- (A e. C -> {x | E.y e. A ph} ~<_ A)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  E.wex 1326  E*wmo 1772  {cab 1871  E.wrex 2106  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  {copab 3395  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993   ~<_ cdom 5424

This theorem is referenced by:  abrexdom2 15740  firnfi 15741

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-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-en 5427  df-dom 5428

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