Theorem infxpidmlem6 7536

Description: Lemma for infxpidm 7543. 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 3456	. . . 4 |- ran U. C = U_g e. C ran g
3	1, 2	eqtr 1494	. . 3 |- B = U_g e. C ran g
4	3	eleq2i 1537	. 2 |- (y e. B <-> y e. U_g e. C ran g)
5		eliun 2567	. 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 955   e. wcel 957  E.wex 979  {cab 1463  E.wrex 1645   (_ wss 2045  (/)c0 2278  U.cuni 2500  U_ciun 2563   class class class wbr 2616  omcom 3128   X. cxp 3165  ran crn 3168  -1-1-onto->wf1o 3178   ~<_ cdom 4362

This theorem is referenced by:  infxpidmlem7 7537  infxpidmlem8 7538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-iun 2565  df-br 2617  df-opab 2664  df-cnv 3183  df-dm 3185  df-rn 3186

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