Theorem dmtpos 8218

Description: The domain of tpos 𝐹 when dom 𝐹 is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
dmtpos	⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)

Proof of Theorem dmtpos

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5700	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
2		ssel 3967	. . . . 5 ⊢ (dom 𝐹 ⊆ (V × V) → (∅ ∈ dom 𝐹 → ∅ ∈ (V × V)))
3	1, 2	mtoi 198	. . . 4 ⊢ (dom 𝐹 ⊆ (V × V) → ¬ ∅ ∈ dom 𝐹)
4		df-rel 5673	. . . 4 ⊢ (Rel dom 𝐹 ↔ dom 𝐹 ⊆ (V × V))
5		reldmtpos 8214	. . . 4 ⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹)
6	3, 4, 5	3imtr4i 292	. . 3 ⊢ (Rel dom 𝐹 → Rel dom tpos 𝐹)
7		relcnv 6093	. . 3 ⊢ Rel ◡dom 𝐹
8	6, 7	jctir 520	. 2 ⊢ (Rel dom 𝐹 → (Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹))
9		vex 3470	. . . . . 6 ⊢ 𝑧 ∈ V
10		brtpos 8215	. . . . . 6 ⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
11	9, 10	mp1i 13	. . . . 5 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧))
12	11	exbidv 1916	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧))
13		opex 5454	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
14	13	eldm 5890	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ∃𝑧⟨𝑥, 𝑦⟩tpos 𝐹𝑧)
15		vex 3470	. . . . . 6 ⊢ 𝑥 ∈ V
16		vex 3470	. . . . . 6 ⊢ 𝑦 ∈ V
17	15, 16	opelcnv 5871	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ⟨𝑦, 𝑥⟩ ∈ dom 𝐹)
18		opex 5454	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
19	18	eldm 5890	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
20	17, 19	bitri 275	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑥⟩𝐹𝑧)
21	12, 14, 20	3bitr4g 314	. . 3 ⊢ (Rel dom 𝐹 → (⟨𝑥, 𝑦⟩ ∈ dom tpos 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡dom 𝐹))
22	21	eqrelrdv2 5785	. 2 ⊢ (((Rel dom tpos 𝐹 ∧ Rel ◡dom 𝐹) ∧ Rel dom 𝐹) → dom tpos 𝐹 = ◡dom 𝐹)
23	8, 22	mpancom 685	1 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3940  ∅c0 4314  ⟨cop 4626   class class class wbr 5138   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  Rel wrel 5671  tpos ctpos 8205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-fv 6541  df-tpos 8206

This theorem is referenced by:  rntpos  8219  dftpos2  8223  dftpos3  8224  tposfn2  8228