Theorem cnvct 8033

Description: If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)

Assertion

Ref	Expression
cnvct	|- ( A ~<_ om -> `' A ~<_ om )

Proof of Theorem cnvct

Step	Hyp	Ref	Expression
1		relcnv 5503	. . . 4 |- Rel `' A
2		ctex 7970	. . . . 5 |- ( A ~<_ om -> A e. _V )
3		cnvexg 7112	. . . . 5 |- ( A e. _V -> `' A e. _V )
4	2, 3	syl 17	. . . 4 |- ( A ~<_ om -> `' A e. _V )
5		cnven 8032	. . . 4 |- ( ( Rel `' A /\ `' A e. _V ) -> `' A ~~ `' `' A )
6	1, 4, 5	sylancr 695	. . 3 |- ( A ~<_ om -> `' A ~~ `' `' A )
7		cnvcnvss 5589	. . . 4 |- `' `' A C_ A
8		ssdomg 8001	. . . 4 |- ( A e. _V -> ( `' `' A C_ A -> `' `' A ~<_ A ) )
9	2, 7, 8	mpisyl 21	. . 3 |- ( A ~<_ om -> `' `' A ~<_ A )
10		endomtr 8014	. . 3 |- ( ( `' A ~~ `' `' A /\ `' `' A ~<_ A ) -> `' A ~<_ A )
11	6, 9, 10	syl2anc 693	. 2 |- ( A ~<_ om -> `' A ~<_ A )
12		domtr 8009	. 2 |- ( ( `' A ~<_ A /\ A ~<_ om ) -> `' A ~<_ om )
13	11, 12	mpancom 703	1 |- ( A ~<_ om -> `' A ~<_ om )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119   omcom 7065   ~~ cen 7952   ~<_ cdom 7953

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-en 7956  df-dom 7957

This theorem is referenced by:  rnct  9347