Theorem disjrnmpt 29398

Description: Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)

Assertion

Ref	Expression
disjrnmpt	|- (Disj x e. A B -> Disj y e. ran ( x e. A |-> B ) y )

Distinct variable groups:   x, A, y   y, B

Allowed substitution hint:   B( x)

Proof of Theorem disjrnmpt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjabrex 29395	. 2 |- (Disj x e. A B -> Disj y e. { z | E. x e. A z = B } y )
2		eqid 2622	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
3	2	rnmpt 5371	. . 3 |- ran ( x e. A |-> B ) = { z | E. x e. A z = B }
4		disjeq1 4627	. . 3 |- ( ran ( x e. A |-> B ) = { z | E. x e. A z = B } -> (Disj y e. ran ( x e. A |-> B ) y <-> Disj y e. { z | E. x e. A z = B } y ) )
5	3, 4	ax-mp 5	. 2 |- (Disj y e. ran ( x e. A |-> B ) y <-> Disj y e. { z | E. x e. A z = B } y )
6	1, 5	sylibr 224	1 |- (Disj x e. A B -> Disj y e. ran ( x e. A |-> B ) y )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   {cab 2608   E.wrex 2913  Disj wdisj 4620   |-> cmpt 4729   ran crn 5115

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-reu 2919  df-rmo 2920  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-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  sigapildsys  30225  ldgenpisyslem1  30226  carsgclctunlem2  30381  pmeasadd  30387