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

Theorem fmpt 5692
 Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fmpt.1 F = (x A C)
Assertion
Ref Expression
fmpt (x A C BF:A–→B)
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   C(x)   F(x)

Proof of Theorem fmpt
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 fmpt.1 . . . 4 F = (x A C)
21fnmpt 5689 . . 3 (x A C BF Fn A)
31rnmpt 5686 . . . 4 ran F = {y x A y = C}
4 r19.29 2754 . . . . . . 7 ((x A C B x A y = C) → x A (C B y = C))
5 eleq1 2413 . . . . . . . . 9 (y = C → (y BC B))
65biimparc 473 . . . . . . . 8 ((C B y = C) → y B)
76rexlimivw 2734 . . . . . . 7 (x A (C B y = C) → y B)
84, 7syl 15 . . . . . 6 ((x A C B x A y = C) → y B)
98ex 423 . . . . 5 (x A C B → (x A y = Cy B))
109abssdv 3340 . . . 4 (x A C B → {y x A y = C} B)
113, 10syl5eqss 3315 . . 3 (x A C B → ran F B)
12 df-f 4791 . . 3 (F:A–→B ↔ (F Fn A ran F B))
132, 11, 12sylanbrc 645 . 2 (x A C BF:A–→B)
141mptpreima 5682 . . . 4 (FB) = {x A C B}
15 fimacnv 5411 . . . 4 (F:A–→B → (FB) = A)
1614, 15syl5reqr 2400 . . 3 (F:A–→BA = {x A C B})
17 rabid2 2788 . . 3 (A = {x A C B} ↔ x A C B)
1816, 17sylib 188 . 2 (F:A–→Bx A C B)
1913, 18impbii 180 1 (x A C BF:A–→B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  {crab 2618   ⊆ wss 3257   “ cima 4722  ◡ccnv 4771  ran crn 4773   Fn wfn 4776  –→wf 4777   ↦ cmpt 5651 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-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-mpt 5652 This theorem is referenced by:  fmpti  5693  fmptd  5694  fmpt2x  5730  enprmaplem5  6080
 Copyright terms: Public domain W3C validator