Theorem rntpos 6501

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

Assertion

Ref	Expression
rntpos	⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Proof of Theorem rntpos

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

Step	Hyp	Ref	Expression
1		vex 2818	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elrn 5005	. . . 4 ⊢ (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3		vex 2818	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3, 1	breldm 4965	. . . . . . . 8 ⊢ (𝑦tpos 𝐹𝑥 → 𝑦 ∈ dom tpos 𝐹)
5		dmtpos 6500	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2304	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (𝑦 ∈ dom tpos 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
7	4, 6	imbitrid 154	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → 𝑦 ∈ ◡dom 𝐹))
8		relcnv 5145	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 4857	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑦 ∈ ◡dom 𝐹) → ∃𝑤∃𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
10	8, 9	mpan 424	. . . . . . 7 ⊢ (𝑦 ∈ ◡dom 𝐹 → ∃𝑤∃𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
11	7, 10	syl6 33	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → ∃𝑤∃𝑧 𝑦 = ⟨𝑤, 𝑧⟩))
12		breq1 4117	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13		vex 2818	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
14		vex 2818	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
15		brtposg 6498	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
16	13, 14, 1, 15	mp3an 1374	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥)
17	12, 16	bitrdi 196	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
18	14, 13	opex 4350	. . . . . . . . 9 ⊢ ⟨𝑧, 𝑤⟩ ∈ V
19	18, 1	brelrn 4995	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩𝐹𝑥 → 𝑥 ∈ ran 𝐹)
20	17, 19	biimtrdi 163	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
21	20	exlimivv 1948	. . . . . 6 ⊢ (∃𝑤∃𝑧 𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
22	11, 21	syli 37	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
23	22	exlimdv 1868	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑦 𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
24	2, 23	biimtrid 152	. . 3 ⊢ (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹 → 𝑥 ∈ ran 𝐹))
25	1	elrn 5005	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
26	3, 1	breldm 4965	. . . . . . 7 ⊢ (𝑦𝐹𝑥 → 𝑦 ∈ dom 𝐹)
27		elrel 4857	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
28	27	ex 115	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑦 ∈ dom 𝐹 → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩))
29	26, 28	syl5 32	. . . . . 6 ⊢ (Rel dom 𝐹 → (𝑦𝐹𝑥 → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩))
30		breq1 4117	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
31	30, 16	bitr4di 198	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
32	13, 14	opex 4350	. . . . . . . . 9 ⊢ ⟨𝑤, 𝑧⟩ ∈ V
33	32, 1	brelrn 4995	. . . . . . . 8 ⊢ (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹)
34	31, 33	biimtrdi 163	. . . . . . 7 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹))
35	34	exlimivv 1948	. . . . . 6 ⊢ (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹))
36	29, 35	syli 37	. . . . 5 ⊢ (Rel dom 𝐹 → (𝑦𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹))
37	36	exlimdv 1868	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑦 𝑦𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹))
38	25, 37	biimtrid 152	. . 3 ⊢ (Rel dom 𝐹 → (𝑥 ∈ ran 𝐹 → 𝑥 ∈ ran tpos 𝐹))
39	24, 38	impbid 129	. 2 ⊢ (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹 ↔ 𝑥 ∈ ran 𝐹))
40	39	eqrdv 2232	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  ⟨cop 3697   class class class wbr 4114  ^◡ccnv 4753  dom cdm 4754  ran crn 4755  Rel wrel 4759  tpos ctpos 6488

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-tpos 6489

This theorem is referenced by:  tposfo2  6511