cnvexg - Intuitionistic Logic Explorer

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Theorem cnvexg 5208

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 5048	. . 3 ⊢ Rel ^◡𝐴
2		relssdmrn 5191	. . 3 ⊢ (Rel ^◡𝐴 → ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴)
4		df-rn 4675	. . . 4 ⊢ ran 𝐴 = dom ^◡𝐴
5		rnexg 4932	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
6	4, 5	eqeltrrid 2284	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom ^◡𝐴 ∈ V)
7		dfdm4 4859	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
8		dmexg 4931	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
9	7, 8	eqeltrrid 2284	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran ^◡𝐴 ∈ V)
10		xpexg 4778	. . 3 ⊢ ((dom ^◡𝐴 ∈ V ∧ ran ^◡𝐴 ∈ V) → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
11	6, 9, 10	syl2anc 411	. 2 ⊢ (𝐴 ∈ 𝑉 → (dom ^◡𝐴 × ran ^◡𝐴) ∈ V)
12		ssexg 4173	. 2 ⊢ ((^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴) ∧ (dom ^◡𝐴 × ran ^◡𝐴) ∈ V) → ^◡𝐴 ∈ V)
13	3, 11, 12	sylancr 414	1 ⊢ (𝐴 ∈ 𝑉 → ^◡𝐴 ∈ V)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2167 Vcvv 2763 ⊆ wss 3157 × cxp 4662 ^◡ccnv 4663 dom cdm 4664 ran crn 4665 Rel wrel 4669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-xp 4670 df-rel 4671 df-cnv 4672 df-dm 4674 df-rn 4675

This theorem is referenced by: cnvex 5209 relcnvexb 5210 cofunex2g 6176 cnvf1o 6292 brtpos2 6318 tposexg 6325 cnven 6876 cnvct 6877 fopwdom 6906 relcnvfi 7016 ennnfonelemim 12666 xpsval 13054 isunitd 13738 znval 14268 znle 14269 znbaslemnn 14271 znleval 14285 pw1nct 15734