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Theorem fmpt 5262
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 A𝐶)
Assertion
Ref Expression
fmpt (x A 𝐶 B𝐹:AB)
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 A𝐶)
21fnmpt 4968 . . 3 (x A 𝐶 B𝐹 Fn A)
31rnmpt 4525 . . . 4 ran 𝐹 = {yx A y = 𝐶}
4 r19.29 2444 . . . . . . 7 ((x A 𝐶 B x A y = 𝐶) → x A (𝐶 B y = 𝐶))
5 eleq1 2097 . . . . . . . . 9 (y = 𝐶 → (y B𝐶 B))
65biimparc 283 . . . . . . . 8 ((𝐶 B y = 𝐶) → y B)
76rexlimivw 2423 . . . . . . 7 (x A (𝐶 B y = 𝐶) → y B)
84, 7syl 14 . . . . . 6 ((x A 𝐶 B x A y = 𝐶) → y B)
98ex 108 . . . . 5 (x A 𝐶 B → (x A y = 𝐶y B))
109abssdv 3008 . . . 4 (x A 𝐶 B → {yx A y = 𝐶} ⊆ B)
113, 10syl5eqss 2983 . . 3 (x A 𝐶 B → ran 𝐹B)
12 df-f 4849 . . 3 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
132, 11, 12sylanbrc 394 . 2 (x A 𝐶 B𝐹:AB)
141mptpreima 4757 . . . 4 (𝐹B) = {x A𝐶 B}
15 fimacnv 5239 . . . 4 (𝐹:AB → (𝐹B) = A)
1614, 15syl5reqr 2084 . . 3 (𝐹:ABA = {x A𝐶 B})
17 rabid2 2480 . . 3 (A = {x A𝐶 B} ↔ x A 𝐶 B)
1816, 17sylib 127 . 2 (𝐹:ABx A 𝐶 B)
1913, 18impbii 117 1 (x A 𝐶 B𝐹:AB)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  {crab 2304  wss 2911  cmpt 3809  ccnv 4287  ran crn 4289  cima 4291   Fn wfn 4840  wf 4841
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-bndl 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-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by:  f1ompt  5263  fmpti  5264  fmptd  5265  rnmptss  5269  f1oresrab  5272  idref  5339  f1mpt  5353  f1stres  5728  f2ndres  5729  fmpt2x  5768  fmpt2co  5779  iunon  5840  dom2lem  6188  uzf  8252
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