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Theorem cnvexg 7631

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 5969	. . 3 ⊢ Rel ^◡𝐴
2		relssdmrn 6123	. . 3 ⊢ (Rel ^◡𝐴 → ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴)
4		df-rn 5568	. . . 4 ⊢ ran 𝐴 = dom ^◡𝐴
5		rnexg 7616	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
6	4, 5	eqeltrrid 2920	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom ^◡𝐴 ∈ V)
7		dfdm4 5766	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
8		dmexg 7615	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
9	7, 8	eqeltrrid 2920	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran ^◡𝐴 ∈ V)
10	6, 9	xpexd 7476	. 2 ⊢ (𝐴 ∈ 𝑉 → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
11		ssexg 5229	. 2 ⊢ ((^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴) ∧ (dom ^◡𝐴 × ran ^◡𝐴) ∈ V) → ^◡𝐴 ∈ V)
12	3, 10, 11	sylancr 589	1 ⊢ (𝐴 ∈ 𝑉 → ^◡𝐴 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 × cxp 5555 ^◡ccnv 5556 dom cdm 5557 ran crn 5558 Rel wrel 5562

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568

This theorem is referenced by: cnvex 7632 relcnvexb 7633 cofunex2g 7653 tposexg 7908 cnven 8587 cnvct 8588 fopwdom 8627 domssex2 8679 domssex 8680 cnvfi 8808 mapfienlem2 8871 wemapwe 9162 hasheqf1oi 13715 brtrclfvcnv 14366 brcnvtrclfvcnv 14367 relexpcnv 14396 relexpnnrn 14406 relexpaddg 14414 imasle 16798 cnvps 17824 gsumvalx 17888 symginv 18532 tposmap 21068 metustel 23162 metustss 23163 metustfbas 23169 metuel2 23177 psmetutop 23179 restmetu 23182 itg2gt0 24363 nlfnval 29660 fnpreimac 30418 ffsrn 30467 tocycfv 30753 eulerpartlemgs2 31640 orvcval 31717 coinfliprv 31742 cossex 35666 cosscnvex 35667 cnvelrels 35737 lkrval 36226 pw2f1o2val 39643 lmhmlnmsplit 39694 cnvcnvintabd 39967 clrellem 39989 relexpaddss 40070 cnvtrclfv 40076 rntrclfvRP 40083 xpexb 40793 sge0f1o 42671 smfco 43084 preimafvelsetpreimafv 43555 fundcmpsurinjlem2 43566