Theorem cnvexg 2669

Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.

Assertion

Ref	Expression
cnvexg	⊢ (A ∈ B → ◡A ∈ V)

Proof of Theorem cnvexg

Step	Hyp	Ref	Expression
1		relcnv 2624	. . 3 ⊢ Rel ◡A
2		relssdr 2668	. . 3 ⊢ (Rel ◡A → ◡A ⊆ (dom ◡A × ran ◡A))
3	1, 2	ax-mp 6	. 2 ⊢ ◡A ⊆ (dom ◡A × ran ◡A)
4		rnexg 2569	. . . . 5 ⊢ (A ∈ B → ran A ∈ V)
5		df-rn 2429	. . . . . 6 ⊢ ran A = dom ◡A
6	5	eleq1i 1152	. . . . 5 ⊢ (ran A ∈ V ↔ dom ◡A ∈ V)
7	4, 6	sylib 173	. . . 4 ⊢ (A ∈ B → dom ◡A ∈ V)
8		dmexg 2551	. . . . 5 ⊢ (A ∈ B → dom A ∈ V)
9		dfdm4 2525	. . . . . 6 ⊢ dom A = ran ◡A
10	9	eleq1i 1152	. . . . 5 ⊢ (dom A ∈ V ↔ ran ◡A ∈ V)
11	8, 10	sylib 173	. . . 4 ⊢ (A ∈ B → ran ◡A ∈ V)
12	7, 11	jca 236	. . 3 ⊢ (A ∈ B → (dom ◡A ∈ V ∧ ran ◡A ∈ V))
13		xpexg 2489	. . 3 ⊢ ((dom ◡A ∈ V ∧ ran ◡A ∈ V) → (dom ◡A × ran ◡A) ∈ V)
14		ssexg 1702	. . 3 ⊢ ((dom ◡A × ran ◡A) ∈ V → (◡A ⊆ (dom ◡A × ran ◡A) → ◡A ∈ V))
15	12, 13, 14	3syl 21	. 2 ⊢ (A ∈ B → (◡A ⊆ (dom ◡A × ran ◡A) → ◡A ∈ V))
16	3, 15	mpi 44	1 ⊢ (A ∈ B → ◡A ∈ V)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   × cxp 2408  ^◡ccnv 2409  dom cdm 2410  ran crn 2411  Rel wrel 2415

This theorem is referenced by:  cnvex 2670  fodom 3613

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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