Theorem rntpos 6225

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 2729	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elrn 4847	. . . 4 ⊢ (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3		vex 2729	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3, 1	breldm 4808	. . . . . . . 8 ⊢ (𝑦tpos 𝐹𝑥 → 𝑦 ∈ dom tpos 𝐹)
5		dmtpos 6224	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2236	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (𝑦 ∈ dom tpos 𝐹 ↔ 𝑦 ∈ ◡dom 𝐹))
7	4, 6	syl5ib 153	. . . . . . 7 ⊢ (Rel dom 𝐹 → (𝑦tpos 𝐹𝑥 → 𝑦 ∈ ◡dom 𝐹))
8		relcnv 4982	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 4706	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑦 ∈ ◡dom 𝐹) → ∃𝑤∃𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
10	8, 9	mpan 421	. . . . . . 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 3985	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13		vex 2729	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
14		vex 2729	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
15		brtposg 6222	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
16	13, 14, 1, 15	mp3an 1327	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥)
17	12, 16	bitrdi 195	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
18	14, 13	opex 4207	. . . . . . . . 9 ⊢ ⟨𝑧, 𝑤⟩ ∈ V
19	18, 1	brelrn 4837	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩𝐹𝑥 → 𝑥 ∈ ran 𝐹)
20	17, 19	syl6bi 162	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
21	20	exlimivv 1884	. . . . . 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 1807	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑦 𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
24	2, 23	syl5bi 151	. . 3 ⊢ (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹 → 𝑥 ∈ ran 𝐹))
25	1	elrn 4847	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
26	3, 1	breldm 4808	. . . . . . 7 ⊢ (𝑦𝐹𝑥 → 𝑦 ∈ dom 𝐹)
27		elrel 4706	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
28	27	ex 114	. . . . . . 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 3985	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
31	30, 16	bitr4di 197	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
32	13, 14	opex 4207	. . . . . . . . 9 ⊢ ⟨𝑤, 𝑧⟩ ∈ V
33	32, 1	brelrn 4837	. . . . . . . 8 ⊢ (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹)
34	31, 33	syl6bi 162	. . . . . . 7 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹))
35	34	exlimivv 1884	. . . . . 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 1807	. . . 4 ⊢ (Rel dom 𝐹 → (∃𝑦 𝑦𝐹𝑥 → 𝑥 ∈ ran tpos 𝐹))
38	25, 37	syl5bi 151	. . 3 ⊢ (Rel dom 𝐹 → (𝑥 ∈ ran 𝐹 → 𝑥 ∈ ran tpos 𝐹))
39	24, 38	impbid 128	. 2 ⊢ (Rel dom 𝐹 → (𝑥 ∈ ran tpos 𝐹 ↔ 𝑥 ∈ ran 𝐹))
40	39	eqrdv 2163	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579   class class class wbr 3982  ^◡ccnv 4603  dom cdm 4604  ran crn 4605  Rel wrel 4609  tpos ctpos 6212

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-tpos 6213

This theorem is referenced by:  tposfo2  6235