cnvexg - Intuitionistic Logic Explorer

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Theorem cnvexg 5168

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 5008	. . 3 ⊢ Rel ^◡𝐴
2		relssdmrn 5151	. . 3 ⊢ (Rel ^◡𝐴 → ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ^◡𝐴 ⊆ (dom ^◡𝐴 × ran ^◡𝐴)
4		df-rn 4639	. . . 4 ⊢ ran 𝐴 = dom ^◡𝐴
5		rnexg 4894	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
6	4, 5	eqeltrrid 2265	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom ^◡𝐴 ∈ V)
7		dfdm4 4821	. . . 4 ⊢ dom 𝐴 = ran ^◡𝐴
8		dmexg 4893	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
9	7, 8	eqeltrrid 2265	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran ^◡𝐴 ∈ V)
10		xpexg 4742	. . 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 4144	. 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 2148 Vcvv 2739 ⊆ wss 3131 × cxp 4626 ^◡ccnv 4627 dom cdm 4628 ran crn 4629 Rel wrel 4633

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-rel 4635 df-cnv 4636 df-dm 4638 df-rn 4639

This theorem is referenced by: cnvex 5169 relcnvexb 5170 cofunex2g 6113 cnvf1o 6228 brtpos2 6254 tposexg 6261 cnven 6810 cnvct 6811 fopwdom 6838 relcnvfi 6942 ennnfonelemim 12427 xpsval 12776 isunitd 13280 pw1nct 14837