Theorem elmapintab 37902

Description: Two ways to say a set is an element of mapped intersection of a class. Here F maps elements of C to elements of |^| { x | ph } or x. (Contributed by RP, 19-Aug-2020.)

Hypotheses

Ref	Expression
elmapintab.1	|- ( A e. B <-> ( A e. C /\ ( F ` A ) e. |^| { x | ph } ) )
elmapintab.2	|- ( A e. E <-> ( A e. C /\ ( F ` A ) e. x ) )

Assertion

Ref	Expression
elmapintab	|- ( A e. B <-> ( A e. C /\ A. x ( ph -> A e. E ) ) )

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

Allowed substitution hints:   ph( x)   B( x)   E( x)

Proof of Theorem elmapintab

Step	Hyp	Ref	Expression
1		elmapintab.1	. 2 |- ( A e. B <-> ( A e. C /\ ( F ` A ) e. |^| { x | ph } ) )
2		fvex 6201	. . . 4 |- ( F ` A ) e. _V
3	2	elintab 4487	. . 3 |- ( ( F ` A ) e. |^| { x | ph } <-> A. x ( ph -> ( F ` A ) e. x ) )
4	3	anbi2i 730	. 2 |- ( ( A e. C /\ ( F ` A ) e. |^| { x | ph } ) <-> ( A e. C /\ A. x ( ph -> ( F ` A ) e. x ) ) )
5		elmapintab.2	. . . . . 6 |- ( A e. E <-> ( A e. C /\ ( F ` A ) e. x ) )
6	5	baibr 945	. . . . 5 |- ( A e. C -> ( ( F ` A ) e. x <-> A e. E ) )
7	6	imbi2d 330	. . . 4 |- ( A e. C -> ( ( ph -> ( F ` A ) e. x ) <-> ( ph -> A e. E ) ) )
8	7	albidv 1849	. . 3 |- ( A e. C -> ( A. x ( ph -> ( F ` A ) e. x ) <-> A. x ( ph -> A e. E ) ) )
9	8	pm5.32i 669	. 2 |- ( ( A e. C /\ A. x ( ph -> ( F ` A ) e. x ) ) <-> ( A e. C /\ A. x ( ph -> A e. E ) ) )
10	1, 4, 9	3bitri 286	1 |- ( A e. B <-> ( A e. C /\ A. x ( ph -> A e. E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   |^|cint 4475   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476  df-iota 5851  df-fv 5896

This theorem is referenced by:  elcnvintab  37908