Theorem fncnvimaeqv 16760

Description: The inverse images of the universal class _V under functions on the universal class _V are the universal class _V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)

Assertion

Ref	Expression
fncnvimaeqv	|- ( F Fn _V -> ( `' F " _V ) = _V )

Proof of Theorem fncnvimaeqv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fncnvima2 6339	. 2 |- ( F Fn _V -> ( `' F " _V ) = { y e. _V | ( F ` y ) e. _V } )
2		fvexd 6203	. . . . 5 |- ( F Fn _V -> ( F ` x ) e. _V )
3	2	biantrud 528	. . . 4 |- ( F Fn _V -> ( x e. _V <-> ( x e. _V /\ ( F ` x ) e. _V ) ) )
4		fveq2 6191	. . . . . 6 |- ( y = x -> ( F ` y ) = ( F ` x ) )
5	4	eleq1d 2686	. . . . 5 |- ( y = x -> ( ( F ` y ) e. _V <-> ( F ` x ) e. _V ) )
6	5	elrab 3363	. . . 4 |- ( x e. { y e. _V | ( F ` y ) e. _V } <-> ( x e. _V /\ ( F ` x ) e. _V ) )
7	3, 6	syl6rbbr 279	. . 3 |- ( F Fn _V -> ( x e. { y e. _V | ( F ` y ) e. _V } <-> x e. _V ) )
8	7	eqrdv 2620	. 2 |- ( F Fn _V -> { y e. _V | ( F ` y ) e. _V } = _V )
9	1, 8	eqtrd 2656	1 |- ( F Fn _V -> ( `' F " _V ) = _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   `'ccnv 5113   "cima 5117   Fn wfn 5883   ` 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-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  bascnvimaeqv  16761