Theorem infxpidmlem6 7557

Description: Lemma for infxpidm 7564. 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 3459	. . . 4 |- ran U. C = U_g e. C ran g
3	1, 2	eqtr 1495	. . 3 |- B = U_g e. C ran g
4	3	eleq2i 1538	. 2 |- (y e. B <-> y e. U_g e. C ran g)
5		eliun 2570	. 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 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646   (_ wss 2047  (/)c0 2280  U.cuni 2503  U_ciun 2566   class class class wbr 2619  omcom 3131   X. cxp 3168  ran crn 3171  -1-1-onto->wf1o 3181   ~<_ cdom 4365

This theorem is referenced by:  infxpidmlem7 7558  infxpidmlem8 7559

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

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