Theorem rnfdmpr 41300

Description: The range of a one-to-one function F of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)

Assertion

Ref	Expression
rnfdmpr	|- ( ( X e. V /\ Y e. W ) -> ( F Fn { X , Y } -> ran F = { ( F ` X ) , ( F ` Y ) } ) )

Proof of Theorem rnfdmpr

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

Step	Hyp	Ref	Expression
1		fnrnfv 6242	. . . 4 |- ( F Fn { X , Y } -> ran F = { x | E. i e. { X , Y } x = ( F ` i ) } )
2	1	adantl 482	. . 3 |- ( ( ( X e. V /\ Y e. W ) /\ F Fn { X , Y } ) -> ran F = { x | E. i e. { X , Y } x = ( F ` i ) } )
3		fveq2 6191	. . . . . . . 8 |- ( i = X -> ( F ` i ) = ( F ` X ) )
4	3	eqeq2d 2632	. . . . . . 7 |- ( i = X -> ( x = ( F ` i ) <-> x = ( F ` X ) ) )
5	4	abbidv 2741	. . . . . 6 |- ( i = X -> { x | x = ( F ` i ) } = { x | x = ( F ` X ) } )
6		fveq2 6191	. . . . . . . 8 |- ( i = Y -> ( F ` i ) = ( F ` Y ) )
7	6	eqeq2d 2632	. . . . . . 7 |- ( i = Y -> ( x = ( F ` i ) <-> x = ( F ` Y ) ) )
8	7	abbidv 2741	. . . . . 6 |- ( i = Y -> { x | x = ( F ` i ) } = { x | x = ( F ` Y ) } )
9	5, 8	iunxprg 4607	. . . . 5 |- ( ( X e. V /\ Y e. W ) -> U_ i e. { X , Y } { x | x = ( F ` i ) } = ( { x | x = ( F ` X ) } u. { x | x = ( F ` Y ) } ) )
10	9	adantr 481	. . . 4 |- ( ( ( X e. V /\ Y e. W ) /\ F Fn { X , Y } ) -> U_ i e. { X , Y } { x | x = ( F ` i ) } = ( { x | x = ( F ` X ) } u. { x | x = ( F ` Y ) } ) )
11		iunab 4566	. . . 4 |- U_ i e. { X , Y } { x | x = ( F ` i ) } = { x | E. i e. { X , Y } x = ( F ` i ) }
12		df-sn 4178	. . . . . . 7 |- { ( F ` X ) } = { x | x = ( F ` X ) }
13	12	eqcomi 2631	. . . . . 6 |- { x | x = ( F ` X ) } = { ( F ` X ) }
14		df-sn 4178	. . . . . . 7 |- { ( F ` Y ) } = { x | x = ( F ` Y ) }
15	14	eqcomi 2631	. . . . . 6 |- { x | x = ( F ` Y ) } = { ( F ` Y ) }
16	13, 15	uneq12i 3765	. . . . 5 |- ( { x | x = ( F ` X ) } u. { x | x = ( F ` Y ) } ) = ( { ( F ` X ) } u. { ( F ` Y ) } )
17		df-pr 4180	. . . . 5 |- { ( F ` X ) , ( F ` Y ) } = ( { ( F ` X ) } u. { ( F ` Y ) } )
18	16, 17	eqtr4i 2647	. . . 4 |- ( { x | x = ( F ` X ) } u. { x | x = ( F ` Y ) } ) = { ( F ` X ) , ( F ` Y ) }
19	10, 11, 18	3eqtr3g 2679	. . 3 |- ( ( ( X e. V /\ Y e. W ) /\ F Fn { X , Y } ) -> { x | E. i e. { X , Y } x = ( F ` i ) } = { ( F ` X ) , ( F ` Y ) } )
20	2, 19	eqtrd 2656	. 2 |- ( ( ( X e. V /\ Y e. W ) /\ F Fn { X , Y } ) -> ran F = { ( F ` X ) , ( F ` Y ) } )
21	20	ex 450	1 |- ( ( X e. V /\ Y e. W ) -> ( F Fn { X , Y } -> ran F = { ( F ` X ) , ( F ` Y ) } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   u. cun 3572   {csn 4177   {cpr 4179   U_ciun 4520   ran crn 5115   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-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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  imarnf1pr  41301