Theorem rntpos 5794

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 2538	. . . . 5 ⊢ x ∈ V
2	1	elrn 4504	. . . 4 ⊢ (x ∈ ran tpos 𝐹 ↔ ∃y ytpos 𝐹x)
3		vex 2538	. . . . . . . . 9 ⊢ y ∈ V
4	3, 1	breldm 4466	. . . . . . . 8 ⊢ (ytpos 𝐹x → y ∈ dom tpos 𝐹)
5		dmtpos 5793	. . . . . . . . 9 ⊢ (Rel dom 𝐹 → dom tpos 𝐹 = ◡dom 𝐹)
6	5	eleq2d 2089	. . . . . . . 8 ⊢ (Rel dom 𝐹 → (y ∈ dom tpos 𝐹 ↔ y ∈ ◡dom 𝐹))
7	4, 6	syl5ib 143	. . . . . . 7 ⊢ (Rel dom 𝐹 → (ytpos 𝐹x → y ∈ ◡dom 𝐹))
8		relcnv 4630	. . . . . . . 8 ⊢ Rel ◡dom 𝐹
9		elrel 4369	. . . . . . . 8 ⊢ ((Rel ◡dom 𝐹 ∧ y ∈ ◡dom 𝐹) → ∃w∃z y = ⟨w, z⟩)
10	8, 9	mpan 402	. . . . . . 7 ⊢ (y ∈ ◡dom 𝐹 → ∃w∃z y = ⟨w, z⟩)
11	7, 10	syl6 29	. . . . . 6 ⊢ (Rel dom 𝐹 → (ytpos 𝐹x → ∃w∃z y = ⟨w, z⟩))
12		breq1 3741	. . . . . . . . 9 ⊢ (y = ⟨w, z⟩ → (ytpos 𝐹x ↔ ⟨w, z⟩tpos 𝐹x))
13		vex 2538	. . . . . . . . . 10 ⊢ w ∈ V
14		vex 2538	. . . . . . . . . 10 ⊢ z ∈ V
15		brtposg 5791	. . . . . . . . . 10 ⊢ ((w ∈ V ∧ z ∈ V ∧ x ∈ V) → (⟨w, z⟩tpos 𝐹x ↔ ⟨z, w⟩𝐹x))
16	13, 14, 1, 15	mp3an 1217	. . . . . . . . 9 ⊢ (⟨w, z⟩tpos 𝐹x ↔ ⟨z, w⟩𝐹x)
17	12, 16	syl6bb 185	. . . . . . . 8 ⊢ (y = ⟨w, z⟩ → (ytpos 𝐹x ↔ ⟨z, w⟩𝐹x))
18	14, 13	opex 3940	. . . . . . . . 9 ⊢ ⟨z, w⟩ ∈ V
19	18, 1	brelrn 4494	. . . . . . . 8 ⊢ (⟨z, w⟩𝐹x → x ∈ ran 𝐹)
20	17, 19	syl6bi 152	. . . . . . 7 ⊢ (y = ⟨w, z⟩ → (ytpos 𝐹x → x ∈ ran 𝐹))
21	20	exlimivv 1758	. . . . . 6 ⊢ (∃w∃z y = ⟨w, z⟩ → (ytpos 𝐹x → x ∈ ran 𝐹))
22	11, 21	syli 33	. . . . 5 ⊢ (Rel dom 𝐹 → (ytpos 𝐹x → x ∈ ran 𝐹))
23	22	exlimdv 1682	. . . 4 ⊢ (Rel dom 𝐹 → (∃y ytpos 𝐹x → x ∈ ran 𝐹))
24	2, 23	syl5bi 141	. . 3 ⊢ (Rel dom 𝐹 → (x ∈ ran tpos 𝐹 → x ∈ ran 𝐹))
25	1	elrn 4504	. . . 4 ⊢ (x ∈ ran 𝐹 ↔ ∃y y𝐹x)
26	3, 1	breldm 4466	. . . . . . 7 ⊢ (y𝐹x → y ∈ dom 𝐹)
27		elrel 4369	. . . . . . . 8 ⊢ ((Rel dom 𝐹 ∧ y ∈ dom 𝐹) → ∃z∃w y = ⟨z, w⟩)
28	27	ex 108	. . . . . . 7 ⊢ (Rel dom 𝐹 → (y ∈ dom 𝐹 → ∃z∃w y = ⟨z, w⟩))
29	26, 28	syl5 28	. . . . . 6 ⊢ (Rel dom 𝐹 → (y𝐹x → ∃z∃w y = ⟨z, w⟩))
30		breq1 3741	. . . . . . . . 9 ⊢ (y = ⟨z, w⟩ → (y𝐹x ↔ ⟨z, w⟩𝐹x))
31	30, 16	syl6bbr 187	. . . . . . . 8 ⊢ (y = ⟨z, w⟩ → (y𝐹x ↔ ⟨w, z⟩tpos 𝐹x))
32	13, 14	opex 3940	. . . . . . . . 9 ⊢ ⟨w, z⟩ ∈ V
33	32, 1	brelrn 4494	. . . . . . . 8 ⊢ (⟨w, z⟩tpos 𝐹x → x ∈ ran tpos 𝐹)
34	31, 33	syl6bi 152	. . . . . . 7 ⊢ (y = ⟨z, w⟩ → (y𝐹x → x ∈ ran tpos 𝐹))
35	34	exlimivv 1758	. . . . . 6 ⊢ (∃z∃w y = ⟨z, w⟩ → (y𝐹x → x ∈ ran tpos 𝐹))
36	29, 35	syli 33	. . . . 5 ⊢ (Rel dom 𝐹 → (y𝐹x → x ∈ ran tpos 𝐹))
37	36	exlimdv 1682	. . . 4 ⊢ (Rel dom 𝐹 → (∃y y𝐹x → x ∈ ran tpos 𝐹))
38	25, 37	syl5bi 141	. . 3 ⊢ (Rel dom 𝐹 → (x ∈ ran 𝐹 → x ∈ ran tpos 𝐹))
39	24, 38	impbid 120	. 2 ⊢ (Rel dom 𝐹 → (x ∈ ran tpos 𝐹 ↔ x ∈ ran 𝐹))
40	39	eqrdv 2020	1 ⊢ (Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535  ⟨cop 3353   class class class wbr 3738  ^◡ccnv 4271  dom cdm 4272  ran crn 4273  Rel wrel 4277  tpos ctpos 5781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-tpos 5782

This theorem is referenced by:  tposfo2  5804