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Theorem ssrnres 5059
 Description: Subset of the range of a restriction. (Contributed by set.mm contributors, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres (B ran (C A) ↔ 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 eqss 3287 . 2 (ran (C ∩ (A × B)) = B ↔ (ran (C ∩ (A × B)) B B ran (C ∩ (A × B))))
2 inss2 3476 . . . . 5 (C ∩ (A × B)) (A × B)
3 rnss 4959 . . . . 5 ((C ∩ (A × B)) (A × B) → ran (C ∩ (A × B)) ran (A × B))
42, 3ax-mp 8 . . . 4 ran (C ∩ (A × B)) ran (A × B)
5 rnxpss 5053 . . . 4 ran (A × B) B
64, 5sstri 3281 . . 3 ran (C ∩ (A × B)) B
76biantrur 492 . 2 (B ran (C ∩ (A × B)) ↔ (ran (C ∩ (A × B)) B B ran (C ∩ (A × B))))
8 ssv 3291 . . . . . . . 8 B V
9 xpss2 4857 . . . . . . . 8 (B V → (A × B) (A × V))
108, 9ax-mp 8 . . . . . . 7 (A × B) (A × V)
11 sslin 3481 . . . . . . 7 ((A × B) (A × V) → (C ∩ (A × B)) (C ∩ (A × V)))
1210, 11ax-mp 8 . . . . . 6 (C ∩ (A × B)) (C ∩ (A × V))
13 df-res 4788 . . . . . 6 (C A) = (C ∩ (A × V))
1412, 13sseqtr4i 3304 . . . . 5 (C ∩ (A × B)) (C A)
15 rnss 4959 . . . . 5 ((C ∩ (A × B)) (C A) → ran (C ∩ (A × B)) ran (C A))
1614, 15ax-mp 8 . . . 4 ran (C ∩ (A × B)) ran (C A)
17 sstr 3280 . . . 4 ((B ran (C ∩ (A × B)) ran (C ∩ (A × B)) ran (C A)) → B ran (C A))
1816, 17mpan2 652 . . 3 (B ran (C ∩ (A × B)) → B ran (C A))
19 ssel 3267 . . . . . . 7 (B ran (C A) → (y By ran (C A)))
20 elrn2 4897 . . . . . . 7 (y ran (C A) ↔ xx, y (C A))
2119, 20syl6ib 217 . . . . . 6 (B ran (C A) → (y Bxx, y (C A)))
2221ancrd 537 . . . . 5 (B ran (C A) → (y B → (xx, y (C A) y B)))
23 elrn2 4897 . . . . . 6 (y ran (C ∩ (A × B)) ↔ xx, y (C ∩ (A × B)))
24 elin 3219 . . . . . . . 8 (x, y (C ∩ (A × B)) ↔ (x, y C x, y (A × B)))
25 opelxp 4811 . . . . . . . . 9 (x, y (A × B) ↔ (x A y B))
2625anbi2i 675 . . . . . . . 8 ((x, y C x, y (A × B)) ↔ (x, y C (x A y B)))
27 opelres 4950 . . . . . . . . . 10 (x, y (C A) ↔ (x, y C x A))
2827anbi1i 676 . . . . . . . . 9 ((x, y (C A) y B) ↔ ((x, y C x A) y B))
29 anass 630 . . . . . . . . 9 (((x, y C x A) y B) ↔ (x, y C (x A y B)))
3028, 29bitr2i 241 . . . . . . . 8 ((x, y C (x A y B)) ↔ (x, y (C A) y B))
3124, 26, 303bitri 262 . . . . . . 7 (x, y (C ∩ (A × B)) ↔ (x, y (C A) y B))
3231exbii 1582 . . . . . 6 (xx, y (C ∩ (A × B)) ↔ x(x, y (C A) y B))
33 19.41v 1901 . . . . . 6 (x(x, y (C A) y B) ↔ (xx, y (C A) y B))
3423, 32, 333bitri 262 . . . . 5 (y ran (C ∩ (A × B)) ↔ (xx, y (C A) y B))
3522, 34syl6ibr 218 . . . 4 (B ran (C A) → (y By ran (C ∩ (A × B))))
3635ssrdv 3278 . . 3 (B ran (C A) → B ran (C ∩ (A × B)))
3718, 36impbii 180 . 2 (B ran (C ∩ (A × B)) ↔ B ran (C A))
381, 7, 373bitr2ri 265 1 (B ran (C A) ↔ ran (C ∩ (A × B)) = B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ⟨cop 4561   × cxp 4770  ran crn 4773   ↾ cres 4774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788 This theorem is referenced by:  rninxp  5060
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