Theorem cnvexg 3519

Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.

Assertion

Ref	Expression
cnvexg	|- (A e. B -> `'A e. V)

Proof of Theorem cnvexg

Step	Hyp	Ref	Expression
1		relcnv 3435	. . 3 |- Rel `'A
2		relssdr 3513	. . 3 |- (Rel `'A -> `'A (_ (dom `' A X. ran `' A))
3	1, 2	ax-mp 7	. 2 |- `'A (_ (dom `' A X. ran `' A)
4		ssexg 2721	. . 3 |- ((`'A (_ (dom `' A X. ran `' A) /\ (dom `' A X. ran `' A) e. V) -> `'A e. V)
5		xpexg 3259	. . . 4 |- ((dom `' A e. V /\ ran `' A e. V) -> (dom `' A X. ran `' A) e. V)
6		rnexg 3359	. . . . 5 |- (A e. B -> ran A e. V)
7		df-rn 3189	. . . . 5 |- ran A = dom `' A
8	6, 7	syl5eqelr 1553	. . . 4 |- (A e. B -> dom `' A e. V)
9		dmexg 3358	. . . . 5 |- (A e. B -> dom A e. V)
10		dfdm4 3305	. . . . 5 |- dom A = ran `' A
11	9, 10	syl5eqelr 1553	. . . 4 |- (A e. B -> ran `' A e. V)
12	5, 8, 11	sylanc 471	. . 3 |- (A e. B -> (dom `' A X. ran `' A) e. V)
13	4, 12	sylan2 451	. 2 |- ((`'A (_ (dom `' A X. ran `' A) /\ A e. B) -> `'A e. V)
14	3, 13	mpan 695	1 |- (A e. B -> `'A e. V)

Syntax hints:   -> wi 3   e. wcel 958  Vcvv 1811   (_ wss 2047   X. cxp 3168  `'ccnv 3169  dom cdm 3170  ran crn 3171  Rel wrel 3175

This theorem is referenced by:  cnvex 3520  relcnvexb 3521  cofunex2g 3581  fodom 4798  mapdiscn 10511  cnvhmpha 10525  cnvhmphb 10526  cnvhmph 10527  hmphsyma 10528

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189

Copyright terms: Public domain