cnvexg - Intuitionistic Logic Explorer

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Theorem cnvexg 5328

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 5168	. . 3 ⊢ Rel ^◡𝐴
2		relssdmrn 5311	. . 3 ⊢ (Rel ^◡𝐴 → ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴)
4		df-rn 4787	. . . 4 ⊢ ran 𝐴 = dom ^◡𝐴
5		rnexg 5049	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
6	4, 5	eqeltrrid 2326	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom ^◡𝐴 ∈ V)
7		dfdm4 4975	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
8		dmexg 5048	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
9	7, 8	eqeltrrid 2326	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran ^◡𝐴 ∈ V)
10		xpexg 4891	. . 3 ⊢ ((dom ^◡𝐴 ∈ V ∧ ran ^◡𝐴 ∈ V) → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
11	6, 9, 10	syl2anc 415	. 2 ⊢ (𝐴 ∈ 𝑉 → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
12		ssexg 4274	. 2 ⊢ ((^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴) ∧ (dom ^◡𝐴 × ran ^◡𝐴) ∈ V) → ^◡𝐴 ∈ V)
13	3, 11, 12	sylancr 418	1 ⊢ (𝐴 ∈ 𝑉 → ^◡𝐴 ∈ V)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2209 Vcvv 2821 ⊆ wss 3220 × cxp 4774 ^◡ccnv 4775 dom cdm 4776 ran crn 4777 Rel wrel 4781

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4132 df-opab 4194 df-xp 4782 df-rel 4783 df-cnv 4784 df-dm 4786 df-rn 4787

This theorem is used by: cnvex 5329 relcnvexb 5330 cofunex2g 6343 cnvf1o 6465 brtpos2 6526 tposexg 6533 cnven 7101 cnvct 7102 fopwdom 7141 relcnvfi 7260 ennnfonelemim 13374 xpsval 14309 isunitd 14523 znval 15081 znle 15082 znbaslemnn 15084 znleval 15098 pw1nct 17250