Theorem rntpos 6154

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 2689	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elrn 4782	. . . 4 ⊢ (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3		vex 2689	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3, 1	breldm 4743	. . . . . . . 8 ⊢ (𝑦tpos 𝐹𝑥 → 𝑦 ∈ dom tpos 𝐹)
5		dmtpos 6153	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2209	. . . . . . . 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 4917	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 4641	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ 𝑦 ∈ ◡dom 𝐹) → ∃𝑤∃𝑧 𝑦 = ⟨𝑤, 𝑧⟩)
10	8, 9	mpan 420	. . . . . . 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 3932	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13		vex 2689	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
14		vex 2689	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
15		brtposg 6151	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
16	13, 14, 1, 15	mp3an 1315	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥)
17	12, 16	syl6bb 195	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
18	14, 13	opex 4151	. . . . . . . . 9 ⊢ ⟨𝑧, 𝑤⟩ ∈ V
19	18, 1	brelrn 4772	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩𝐹𝑥 → 𝑥 ∈ ran 𝐹)
20	17, 19	syl6bi 162	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
21	20	exlimivv 1868	. . . . . 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 1791	. . . 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 4782	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
26	3, 1	breldm 4743	. . . . . . 7 ⊢ (𝑦𝐹𝑥 → 𝑦 ∈ dom 𝐹)
27		elrel 4641	. . . . . . . 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 3932	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
31	30, 16	syl6bbr 197	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
32	13, 14	opex 4151	. . . . . . . . 9 ⊢ ⟨𝑤, 𝑧⟩ ∈ V
33	32, 1	brelrn 4772	. . . . . . . 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 1868	. . . . . 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 1791	. . . 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 2137	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  ⟨cop 3530   class class class wbr 3929  ^◡ccnv 4538  dom cdm 4539  ran crn 4540  Rel wrel 4544  tpos ctpos 6141

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-tpos 6142

This theorem is referenced by:  tposfo2  6164