Theorem ac7g 3570

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(s):   R,f

Proof of Theorem ac7g

Step	Hyp	Ref	Expression
1		sseq2 1522	. . . 4 |- (x = R -> (f (_ x <-> f (_ R))
2		dmeq 2531	. . . . 5 |- (x = R -> dom x = dom R)
3		fneq2 2719	. . . . 5 |- (dom x = dom R -> (f Fn dom x <-> f Fn dom R))
4	2, 3	syl 12	. . . 4 |- (x = R -> (f Fn dom x <-> f Fn dom R))
5	1, 4	anbi12d 476	. . 3 |- (x = R -> ((f (_ x /\ f Fn dom x) <-> (f (_ R /\ f Fn dom R)))
6	5	biexdv 936	. 2 |- (x = R -> (E.f(f (_ x /\ f Fn dom x) <-> E.f(f (_ R /\ f Fn dom R)))
7		ac7 3569	. 2 |- E.f(f (_ x /\ f Fn dom x)
8	6, 7	vtoclg 1383	1 |- (R e. A -> E.f(f (_ R /\ f Fn dom R))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092   (_ wss 1487  dom cdm 2410   Fn wfn 2417

This theorem is referenced by:  fodom 3613  infmap2lem2 4952

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-fr 2169  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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