Theorem cnvexg 3511

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 3427	. . 3 |- Rel `'A
2		relssdr 3505	. . 3 |- (Rel `'A -> `'A (_ (dom `' A X. ran `' A))
3	1, 2	ax-mp 7	. 2 |- `'A (_ (dom `' A X. ran `' A)
4		ssexg 2716	. . 3 |- ((`'A (_ (dom `' A X. ran `' A) /\ (dom `' A X. ran `' A) e. V) -> `'A e. V)
5		xpexg 3254	. . . 4 |- ((dom `' A e. V /\ ran `' A e. V) -> (dom `' A X. ran `' A) e. V)
6		rnexg 3353	. . . . 5 |- (A e. B -> ran A e. V)
7		df-rn 3184	. . . . 5 |- ran A = dom `' A
8	6, 7	syl5eqelr 1550	. . . 4 |- (A e. B -> dom `' A e. V)
9		dmexg 3352	. . . . 5 |- (A e. B -> dom A e. V)
10		dfdm4 3300	. . . . 5 |- dom A = ran `' A
11	9, 10	syl5eqelr 1550	. . . 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 694	1 |- (A e. B -> `'A e. V)

Syntax hints:   -> wi 3   e. wcel 956  Vcvv 1807   (_ wss 2043   X. cxp 3163  `'ccnv 3164  dom cdm 3165  ran crn 3166  Rel wrel 3170

This theorem is referenced by:  cnvex 3512  relcnvexb 3513  cofunex2g 3573  fodom 4778  mapdiscn 10434  cnvhmpha 10448  cnvhmphb 10449  cnvhmph 10450  hmphsyma 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184

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