Theorem abnexg 4406

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 6066. 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 4408 and pwnex 4409 respectively, proved from abnex 4407, which is a consequence of abnexg 4406 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 4399	. 2 |- ( { y | E. x e. A y = F } e. W -> U. { y | E. x e. A y = F } e. _V )
2		simpl 108	. . . . 5 |- ( ( F e. V /\ x e. F ) -> F e. V )
3	2	ralimi 2520	. . . 4 |- ( A. x e. A ( F e. V /\ x e. F ) -> A. x e. A F e. V )
4		dfiun2g 3881	. . . . . 6 |- ( A. x e. A F e. V -> U_ x e. A F = U. { y | E. x e. A y = F } )
5	4	eleq1d 2226	. . . . 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 157	. . . 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 109	. . . . 5 |- ( ( F e. V /\ x e. F ) -> x e. F )
9	8	ralimi 2520	. . . 4 |- ( A. x e. A ( F e. V /\ x e. F ) -> A. x e. A x e. F )
10		iunid 3904	. . . . 5 |- U_ x e. A { x } = A
11		snssi 3700	. . . . . . 7 |- ( x e. F -> { x } C_ F )
12	11	ralimi 2520	. . . . . 6 |- ( A. x e. A x e. F -> A. x e. A { x } C_ F )
13		ss2iun 3864	. . . . . 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 3175	. . . 4 |- ( A. x e. A x e. F -> A C_ U_ x e. A F )
16		ssexg 4103	. . . . 5 |- ( ( A C_ U_ x e. A F /\ U_ x e. A F e. _V ) -> A e. _V )
17	16	ex 114	. . . 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 103   = wceq 1335   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   _Vcvv 2712   C_ wss 3102   {csn 3560   U.cuni 3772   U_ciun 3849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-sn 3566  df-uni 3773  df-iun 3851

This theorem is referenced by:  abnex  4407