Theorem rntpos 6414

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 2802	. . . . 5 ⊢ 𝑥 ∈ V
2	1	elrn 4970	. . . 4 ⊢ (𝑥 ∈ ran tpos 𝐹 ↔ ∃𝑦 𝑦tpos 𝐹𝑥)
3		vex 2802	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3, 1	breldm 4930	. . . . . . . 8 ⊢ (𝑦tpos 𝐹𝑥 → 𝑦 ∈ dom tpos 𝐹)
5		dmtpos 6413	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2299	. . . . . . . 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 5109	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 4823	. . . . . . . 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 4086	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 ↔ ⟨𝑤, 𝑧⟩tpos 𝐹𝑥))
13		vex 2802	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
14		vex 2802	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
15		brtposg 6411	. . . . . . . . . 10 ⊢ ((𝑤 ∈ V ∧ 𝑧 ∈ V ∧ 𝑥 ∈ V) → (⟨𝑤, 𝑧⟩tpos 𝐹𝑥 ↔ ⟨𝑧, 𝑤⟩𝐹𝑥))
16	13, 14, 1, 15	mp3an 1371	. . . . . . . . 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 4316	. . . . . . . . 9 ⊢ ⟨𝑧, 𝑤⟩ ∈ V
19	18, 1	brelrn 4960	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩𝐹𝑥 → 𝑥 ∈ ran 𝐹)
20	17, 19	biimtrdi 163	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 𝑧⟩ → (𝑦tpos 𝐹𝑥 → 𝑥 ∈ ran 𝐹))
21	20	exlimivv 1943	. . . . . 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 1865	. . . 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 4970	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 𝑦𝐹𝑥)
26	3, 1	breldm 4930	. . . . . . 7 ⊢ (𝑦𝐹𝑥 → 𝑦 ∈ dom 𝐹)
27		elrel 4823	. . . . . . . 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 4086	. . . . . . . . 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 4316	. . . . . . . . 9 ⊢ ⟨𝑤, 𝑧⟩ ∈ V
33	32, 1	brelrn 4960	. . . . . . . 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 1943	. . . . . 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 1865	. . . 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 2227	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669   class class class wbr 4083  ^◡ccnv 4719  dom cdm 4720  ran crn 4721  Rel wrel 4725  tpos ctpos 6401

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-fv 5329  df-tpos 6402

This theorem is referenced by:  tposfo2  6424