Theorem infxpidmlem6 7558

Description: Lemma for infxpidm 7565. A simple but frequently used fact.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem6.2	|- B = ran U. C

Assertion

Ref	Expression
infxpidmlem6	|- (y e. B <-> E.g e. C y e. ran g)

Distinct variable groups:   y,f,g,t,A   y,B,f,g,t   y,C,f,g,t   y,H,g

Proof of Theorem infxpidmlem6

Step	Hyp	Ref	Expression
1		infxpidmlem6.2	. . . 4 |- B = ran U. C
2		rnuni 3465	. . . 4 |- ran U. C = U_g e. C ran g
3	1, 2	eqtr 1498	. . 3 |- B = U_g e. C ran g
4	3	eleq2i 1541	. 2 |- (y e. B <-> y e. U_g e. C ran g)
5		eliun 2574	. 2 |- (y e. U_g e. C ran g <-> E.g e. C y e. ran g)
6	4, 5	bitr 173	1 |- (y e. B <-> E.g e. C y e. ran g)

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649   (_ wss 2050  (/)c0 2283  U.cuni 2507  U_ciun 2570   class class class wbr 2624  omcom 3137   X. cxp 3174  ran crn 3177  -1-1-onto->wf1o 3187   ~<_ cdom 4371

This theorem is referenced by:  infxpidmlem7 7559  infxpidmlem8 7560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195

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