New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  imainss GIF version

Theorem imainss 5042
 Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by set.mm contributors, 11-Aug-2004.)
Assertion
Ref Expression
imainss ((RA) ∩ B) (R “ (A ∩ (RB)))

Proof of Theorem imainss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 730 . . . . . 6 (((x A xRy) y B) → x A)
2 brcnv 4892 . . . . . . . . 9 (yRxxRy)
3 19.8a 1756 . . . . . . . . 9 ((y B yRx) → y(y B yRx))
42, 3sylan2br 462 . . . . . . . 8 ((y B xRy) → y(y B yRx))
54ancoms 439 . . . . . . 7 ((xRy y B) → y(y B yRx))
65adantll 694 . . . . . 6 (((x A xRy) y B) → y(y B yRx))
71, 6jca 518 . . . . 5 (((x A xRy) y B) → (x A y(y B yRx)))
8 simplr 731 . . . . 5 (((x A xRy) y B) → xRy)
9 elin 3219 . . . . . . 7 (x (A ∩ (RB)) ↔ (x A x (RB)))
10 elima2 4755 . . . . . . . 8 (x (RB) ↔ y(y B yRx))
1110anbi2i 675 . . . . . . 7 ((x A x (RB)) ↔ (x A y(y B yRx)))
129, 11bitri 240 . . . . . 6 (x (A ∩ (RB)) ↔ (x A y(y B yRx)))
1312anbi1i 676 . . . . 5 ((x (A ∩ (RB)) xRy) ↔ ((x A y(y B yRx)) xRy))
147, 8, 13sylanbrc 645 . . . 4 (((x A xRy) y B) → (x (A ∩ (RB)) xRy))
1514eximi 1576 . . 3 (x((x A xRy) y B) → x(x (A ∩ (RB)) xRy))
16 elima2 4755 . . . . 5 (y (RA) ↔ x(x A xRy))
1716anbi1i 676 . . . 4 ((y (RA) y B) ↔ (x(x A xRy) y B))
18 elin 3219 . . . 4 (y ((RA) ∩ B) ↔ (y (RA) y B))
19 19.41v 1901 . . . 4 (x((x A xRy) y B) ↔ (x(x A xRy) y B))
2017, 18, 193bitr4i 268 . . 3 (y ((RA) ∩ B) ↔ x((x A xRy) y B))
21 elima2 4755 . . 3 (y (R “ (A ∩ (RB))) ↔ x(x (A ∩ (RB)) xRy))
2215, 20, 213imtr4i 257 . 2 (y ((RA) ∩ B) → y (R “ (A ∩ (RB))))
2322ssriv 3277 1 ((RA) ∩ B) (R “ (A ∩ (RB)))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257   class class class wbr 4639   “ cima 4722  ◡ccnv 4771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-cnv 4785 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator