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Theorem cnvexg 7005

Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

**Assertion**
Ref	Expression
cnvexg	⊢ (𝐴 ∈ 𝑉 → ^◡𝐴 ∈ V)

**Proof of Theorem cnvexg**
Step	Hyp	Ref	Expression
1		relcnv 5422	. . 3 ⊢ Rel ^◡𝐴
2		relssdmrn 5573	. . 3 ⊢ (Rel ^◡𝐴 → ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴)
4		df-rn 5049	. . . 4 ⊢ ran 𝐴 = dom ^◡𝐴
5		rnexg 6990	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
6	4, 5	syl5eqelr 2693	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom ^◡𝐴 ∈ V)
7		dfdm4 5238	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
8		dmexg 6989	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
9	7, 8	syl5eqelr 2693	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran ^◡𝐴 ∈ V)
10		xpexg 6858	. . 3 ⊢ ((dom ^◡𝐴 ∈ V ∧ ran ^◡𝐴 ∈ V) → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
11	6, 9, 10	syl2anc 691	. 2 ⊢ (𝐴 ∈ 𝑉 → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
12		ssexg 4732	. 2 ⊢ ((^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴) ∧ (dom ^◡𝐴 × ran ^◡𝐴) ∈ V) → ^◡𝐴 ∈ V)
13	3, 11, 12	sylancr 694	1 ⊢ (𝐴 ∈ 𝑉 → ^◡𝐴 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 × cxp 5036 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 Rel wrel 5043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049

This theorem is referenced by: cnvex 7006 relcnvexb 7007 cofunex2g 7024 tposexg 7253 cnven 7918 fopwdom 7953 domssex2 8005 domssex 8006 cnvfi 8131 mapfienlem2 8194 wemapwe 8477 hasheqf1oi 13002 hasheqf1oiOLD 13003 brtrclfvcnv 13593 brcnvtrclfvcnv 13594 relexpcnv 13623 relexpnnrn 13633 relexpaddg 13641 imasle 16006 cnvps 17035 gsumvalx 17093 symginv 17645 tposmap 20082 metustel 22165 metustss 22166 metustfbas 22172 metuel2 22180 psmetutop 22182 restmetu 22185 itg2gt0 23333 nlfnval 28124 cnvct 28878 ffsrn 28892 eulerpartlemgs2 29769 orvcval 29846 coinfliprv 29871 lkrval 33393 pw2f1o2val 36624 lmhmlnmsplit 36675 cnvcnvintabd 36925 clrellem 36948 relexpaddss 37029 cnvtrclfv 37035 rntrclfvRP 37042 xpexb 37679 sge0f1o 39275 smfco 39687