Theorem abnexg 4477

Description: Sufficient condition for a class abstraction to be a proper class. The class F can be thought of as an expression in x and the abstraction appearing in the statement as the class of values F as x varies through A. Assuming the antecedents, if that class is a set, then so is the "domain" A. The converse holds without antecedent, see abrexexg 6170. Note that the second antecedent A. x e. A x e. F cannot be translated to A C_ F since F may depend on x. In applications, one may take F = { x } or F = ~P x (see snnex 4479 and pwnex 4480 respectively, proved from abnex 4478, which is a consequence of abnexg 4477 with A = _V). (Contributed by BJ, 2-Dec-2021.)

Assertion

Ref	Expression
abnexg	|- ( A. x e. A ( F e. V /\ x e. F ) -> ( { y | E. x e. A y = F } e. W -> A e. _V ) )

Distinct variable groups:   x, A, y   y, F

Allowed substitution hints:   F( x)   V( x, y)   W( x, y)

Proof of Theorem abnexg

Step	Hyp	Ref	Expression
1		uniexg 4470	. 2 |- ( { y | E. x e. A y = F } e. W -> U. { y | E. x e. A y = F } e. _V )
2		simpl 109	. . . . 5 |- ( ( F e. V /\ x e. F ) -> F e. V )
3	2	ralimi 2557	. . . 4 |- ( A. x e. A ( F e. V /\ x e. F ) -> A. x e. A F e. V )
4		dfiun2g 3944	. . . . . 6 |- ( A. x e. A F e. V -> U_ x e. A F = U. { y | E. x e. A y = F } )
5	4	eleq1d 2262	. . . . 5 |- ( A. x e. A F e. V -> ( U_ x e. A F e. _V <-> U. { y | E. x e. A y = F } e. _V ) )
6	5	biimprd 158	. . . 4 |- ( A. x e. A F e. V -> ( U. { y | E. x e. A y = F } e. _V -> U_ x e. A F e. _V ) )
7	3, 6	syl 14	. . 3 |- ( A. x e. A ( F e. V /\ x e. F ) -> ( U. { y | E. x e. A y = F } e. _V -> U_ x e. A F e. _V ) )
8		simpr 110	. . . . 5 |- ( ( F e. V /\ x e. F ) -> x e. F )
9	8	ralimi 2557	. . . 4 |- ( A. x e. A ( F e. V /\ x e. F ) -> A. x e. A x e. F )
10		iunid 3968	. . . . 5 |- U_ x e. A { x } = A
11		snssi 3762	. . . . . . 7 |- ( x e. F -> { x } C_ F )
12	11	ralimi 2557	. . . . . 6 |- ( A. x e. A x e. F -> A. x e. A { x } C_ F )
13		ss2iun 3927	. . . . . 6 |- ( A. x e. A { x } C_ F -> U_ x e. A { x } C_ U_ x e. A F )
14	12, 13	syl 14	. . . . 5 |- ( A. x e. A x e. F -> U_ x e. A { x } C_ U_ x e. A F )
15	10, 14	eqsstrrid 3226	. . . 4 |- ( A. x e. A x e. F -> A C_ U_ x e. A F )
16		ssexg 4168	. . . . 5 |- ( ( A C_ U_ x e. A F /\ U_ x e. A F e. _V ) -> A e. _V )
17	16	ex 115	. . . 4 |- ( A C_ U_ x e. A F -> ( U_ x e. A F e. _V -> A e. _V ) )
18	9, 15, 17	3syl 17	. . 3 |- ( A. x e. A ( F e. V /\ x e. F ) -> ( U_ x e. A F e. _V -> A e. _V ) )
19	7, 18	syld 45	. 2 |- ( A. x e. A ( F e. V /\ x e. F ) -> ( U. { y | E. x e. A y = F } e. _V -> A e. _V ) )
20	1, 19	syl5 32	1 |- ( A. x e. A ( F e. V /\ x e. F ) -> ( { y | E. x e. A y = F } e. W -> A e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   C_ wss 3153   {csn 3618   U.cuni 3835   U_ciun 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-sn 3624  df-uni 3836  df-iun 3914

This theorem is referenced by:  abnex  4478