ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssrnres Structured version   GIF version

Theorem ssrnres 4706
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres (B ⊆ ran (𝐶A) ↔ ran (𝐶 ∩ (A × B)) = B)

Proof of Theorem ssrnres
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3152 . . . . 5 (𝐶 ∩ (A × B)) ⊆ (A × B)
2 rnss 4507 . . . . 5 ((𝐶 ∩ (A × B)) ⊆ (A × B) → ran (𝐶 ∩ (A × B)) ⊆ ran (A × B))
31, 2ax-mp 7 . . . 4 ran (𝐶 ∩ (A × B)) ⊆ ran (A × B)
4 rnxpss 4697 . . . 4 ran (A × B) ⊆ B
53, 4sstri 2948 . . 3 ran (𝐶 ∩ (A × B)) ⊆ B
6 eqss 2954 . . 3 (ran (𝐶 ∩ (A × B)) = B ↔ (ran (𝐶 ∩ (A × B)) ⊆ B B ⊆ ran (𝐶 ∩ (A × B))))
75, 6mpbiran 846 . 2 (ran (𝐶 ∩ (A × B)) = BB ⊆ ran (𝐶 ∩ (A × B)))
8 ssid 2958 . . . . . . . 8 AA
9 ssv 2959 . . . . . . . 8 B ⊆ V
10 xpss12 4388 . . . . . . . 8 ((AA B ⊆ V) → (A × B) ⊆ (A × V))
118, 9, 10mp2an 402 . . . . . . 7 (A × B) ⊆ (A × V)
12 sslin 3157 . . . . . . 7 ((A × B) ⊆ (A × V) → (𝐶 ∩ (A × B)) ⊆ (𝐶 ∩ (A × V)))
1311, 12ax-mp 7 . . . . . 6 (𝐶 ∩ (A × B)) ⊆ (𝐶 ∩ (A × V))
14 df-res 4300 . . . . . 6 (𝐶A) = (𝐶 ∩ (A × V))
1513, 14sseqtr4i 2972 . . . . 5 (𝐶 ∩ (A × B)) ⊆ (𝐶A)
16 rnss 4507 . . . . 5 ((𝐶 ∩ (A × B)) ⊆ (𝐶A) → ran (𝐶 ∩ (A × B)) ⊆ ran (𝐶A))
1715, 16ax-mp 7 . . . 4 ran (𝐶 ∩ (A × B)) ⊆ ran (𝐶A)
18 sstr 2947 . . . 4 ((B ⊆ ran (𝐶 ∩ (A × B)) ran (𝐶 ∩ (A × B)) ⊆ ran (𝐶A)) → B ⊆ ran (𝐶A))
1917, 18mpan2 401 . . 3 (B ⊆ ran (𝐶 ∩ (A × B)) → B ⊆ ran (𝐶A))
20 ssel 2933 . . . . . . 7 (B ⊆ ran (𝐶A) → (y By ran (𝐶A)))
21 vex 2554 . . . . . . . 8 y V
2221elrn2 4519 . . . . . . 7 (y ran (𝐶A) ↔ xx, y (𝐶A))
2320, 22syl6ib 150 . . . . . 6 (B ⊆ ran (𝐶A) → (y Bxx, y (𝐶A)))
2423ancrd 309 . . . . 5 (B ⊆ ran (𝐶A) → (y B → (xx, y (𝐶A) y B)))
2521elrn2 4519 . . . . . 6 (y ran (𝐶 ∩ (A × B)) ↔ xx, y (𝐶 ∩ (A × B)))
26 elin 3120 . . . . . . . 8 (⟨x, y (𝐶 ∩ (A × B)) ↔ (⟨x, y 𝐶 x, y (A × B)))
27 opelxp 4317 . . . . . . . . 9 (⟨x, y (A × B) ↔ (x A y B))
2827anbi2i 430 . . . . . . . 8 ((⟨x, y 𝐶 x, y (A × B)) ↔ (⟨x, y 𝐶 (x A y B)))
2921opelres 4560 . . . . . . . . . 10 (⟨x, y (𝐶A) ↔ (⟨x, y 𝐶 x A))
3029anbi1i 431 . . . . . . . . 9 ((⟨x, y (𝐶A) y B) ↔ ((⟨x, y 𝐶 x A) y B))
31 anass 381 . . . . . . . . 9 (((⟨x, y 𝐶 x A) y B) ↔ (⟨x, y 𝐶 (x A y B)))
3230, 31bitr2i 174 . . . . . . . 8 ((⟨x, y 𝐶 (x A y B)) ↔ (⟨x, y (𝐶A) y B))
3326, 28, 323bitri 195 . . . . . . 7 (⟨x, y (𝐶 ∩ (A × B)) ↔ (⟨x, y (𝐶A) y B))
3433exbii 1493 . . . . . 6 (xx, y (𝐶 ∩ (A × B)) ↔ x(⟨x, y (𝐶A) y B))
35 19.41v 1779 . . . . . 6 (x(⟨x, y (𝐶A) y B) ↔ (xx, y (𝐶A) y B))
3625, 34, 353bitri 195 . . . . 5 (y ran (𝐶 ∩ (A × B)) ↔ (xx, y (𝐶A) y B))
3724, 36syl6ibr 151 . . . 4 (B ⊆ ran (𝐶A) → (y By ran (𝐶 ∩ (A × B))))
3837ssrdv 2945 . . 3 (B ⊆ ran (𝐶A) → B ⊆ ran (𝐶 ∩ (A × B)))
3919, 38impbii 117 . 2 (B ⊆ ran (𝐶 ∩ (A × B)) ↔ B ⊆ ran (𝐶A))
407, 39bitr2i 174 1 (B ⊆ ran (𝐶A) ↔ ran (𝐶 ∩ (A × B)) = B)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cin 2910  wss 2911  cop 3370   × cxp 4286  ran crn 4289  cres 4290
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by:  rninxp  4707
  Copyright terms: Public domain W3C validator