Theorem dfcnv2 29476

Description: Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)

Assertion

Ref	Expression
dfcnv2	|- ( ran R C_ A -> `' R = U_ x e. A ( { x } X. ( `' R " { x } ) ) )

Distinct variable groups:   x, A   x, R

Proof of Theorem dfcnv2

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

Step	Hyp	Ref	Expression
1		relcnv 5503	. 2 |- Rel `' R
2		relxp 5227	. . . 4 |- Rel ( { x } X. ( `' R " { x } ) )
3	2	rgenw 2924	. . 3 |- A. x e. A Rel ( { x } X. ( `' R " { x } ) )
4		reliun 5239	. . 3 |- ( Rel U_ x e. A ( { x } X. ( `' R " { x } ) ) <-> A. x e. A Rel ( { x } X. ( `' R " { x } ) ) )
5	3, 4	mpbir 221	. 2 |- Rel U_ x e. A ( { x } X. ( `' R " { x } ) )
6		vex 3203	. . . . . . . . 9 |- z e. _V
7		vex 3203	. . . . . . . . 9 |- y e. _V
8	6, 7	opeldm 5328	. . . . . . . 8 |- ( <. z , y >. e. `' R -> z e. dom `' R )
9		df-rn 5125	. . . . . . . 8 |- ran R = dom `' R
10	8, 9	syl6eleqr 2712	. . . . . . 7 |- ( <. z , y >. e. `' R -> z e. ran R )
11		ssel2 3598	. . . . . . 7 |- ( ( ran R C_ A /\ z e. ran R ) -> z e. A )
12	10, 11	sylan2 491	. . . . . 6 |- ( ( ran R C_ A /\ <. z , y >. e. `' R ) -> z e. A )
13	12	ex 450	. . . . 5 |- ( ran R C_ A -> ( <. z , y >. e. `' R -> z e. A ) )
14	13	pm4.71rd 667	. . . 4 |- ( ran R C_ A -> ( <. z , y >. e. `' R <-> ( z e. A /\ <. z , y >. e. `' R ) ) )
15	6, 7	elimasn 5490	. . . . 5 |- ( y e. ( `' R " { z } ) <-> <. z , y >. e. `' R )
16	15	anbi2i 730	. . . 4 |- ( ( z e. A /\ y e. ( `' R " { z } ) ) <-> ( z e. A /\ <. z , y >. e. `' R ) )
17	14, 16	syl6bbr 278	. . 3 |- ( ran R C_ A -> ( <. z , y >. e. `' R <-> ( z e. A /\ y e. ( `' R " { z } ) ) ) )
18		sneq 4187	. . . . 5 |- ( x = z -> { x } = { z } )
19	18	imaeq2d 5466	. . . 4 |- ( x = z -> ( `' R " { x } ) = ( `' R " { z } ) )
20	19	opeliunxp2 5260	. . 3 |- ( <. z , y >. e. U_ x e. A ( { x } X. ( `' R " { x } ) ) <-> ( z e. A /\ y e. ( `' R " { z } ) ) )
21	17, 20	syl6bbr 278	. 2 |- ( ran R C_ A -> ( <. z , y >. e. `' R <-> <. z , y >. e. U_ x e. A ( { x } X. ( `' R " { x } ) ) ) )
22	1, 5, 21	eqrelrdv 5216	1 |- ( ran R C_ A -> `' R = U_ x e. A ( { x } X. ( `' R " { x } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119

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-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  gsummpt2co  29780