Theorem cnvexg 5148

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 4989	. . 3 ⊢ Rel ◡𝐴
2		relssdmrn 5131	. . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)
4		df-rn 4622	. . . 4 ⊢ ran 𝐴 = dom ◡𝐴
5		rnexg 4876	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
6	4, 5	eqeltrrid 2258	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom ◡𝐴 ∈ V)
7		dfdm4 4803	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
8		dmexg 4875	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
9	7, 8	eqeltrrid 2258	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran ◡𝐴 ∈ V)
10		xpexg 4725	. . 3 ⊢ ((dom ◡𝐴 ∈ V ∧ ran ◡𝐴 ∈ V) → (dom ◡𝐴 × ran ◡𝐴) ∈ V)
11	6, 9, 10	syl2anc 409	. 2 ⊢ (𝐴 ∈ 𝑉 → (dom ◡𝐴 × ran ◡𝐴) ∈ V)
12		ssexg 4128	. 2 ⊢ ((◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴) ∧ (dom ◡𝐴 × ran ◡𝐴) ∈ V) → ◡𝐴 ∈ V)
13	3, 11, 12	sylancr 412	1 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121   × cxp 4609  ^◡ccnv 4610  dom cdm 4611  ran crn 4612  Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  cnvex  5149  relcnvexb  5150  cofunex2g  6089  cnvf1o  6204  brtpos2  6230  tposexg  6237  cnven  6786  cnvct  6787  fopwdom  6814  relcnvfi  6918  ennnfonelemim  12379  pw1nct  14036