Theorem ac7g 4759

Description: An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49.

Assertion

Ref	Expression
ac7g	|- (R e. A -> E.f(f (_ R /\ f Fn dom R))

Distinct variable group:   R,f

Proof of Theorem ac7g

Step	Hyp	Ref	Expression
1		sseq2 2086	. . . 4 |- (x = R -> (f (_ x <-> f (_ R))
2		dmeq 3317	. . . . 5 |- (x = R -> dom x = dom R)
3		fneq2 3589	. . . . 5 |- (dom x = dom R -> (f Fn dom x <-> f Fn dom R))
4	2, 3	syl 10	. . . 4 |- (x = R -> (f Fn dom x <-> f Fn dom R))
5	1, 4	anbi12d 630	. . 3 |- (x = R -> ((f (_ x /\ f Fn dom x) <-> (f (_ R /\ f Fn dom R)))
6	5	exbidv 1281	. 2 |- (x = R -> (E.f(f (_ x /\ f Fn dom x) <-> E.f(f (_ R /\ f Fn dom R)))
7		ac7 4758	. 2 |- E.f(f (_ x /\ f Fn dom x)
8	6, 7	vtoclg 1850	1 |- (R e. A -> E.f(f (_ R /\ f Fn dom R))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   (_ wss 2050  dom cdm 3176   Fn wfn 3183

This theorem is referenced by:  fodom 4808  infmap2lem2 7582

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-9 967  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-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

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-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  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-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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