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Theorem fmpt 5244
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 4951 . . 3 (x A 𝐶 B𝐹 Fn A)
31rnmpt 4509 . . . 4 ran 𝐹 = {yx A y = 𝐶}
4 r19.29 2428 . . . . . . 7 ((x A 𝐶 B x A y = 𝐶) → x A (𝐶 B y = 𝐶))
5 eleq1 2082 . . . . . . . . 9 (y = 𝐶 → (y B𝐶 B))
65biimparc 283 . . . . . . . 8 ((𝐶 B y = 𝐶) → y B)
76rexlimivw 2407 . . . . . . 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 2991 . . . 4 (x A 𝐶 B → {yx A y = 𝐶} ⊆ B)
113, 10syl5eqss 2966 . . 3 (x A 𝐶 B → ran 𝐹B)
12 df-f 4833 . . 3 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
132, 11, 12sylanbrc 396 . 2 (x A 𝐶 B𝐹:AB)
141mptpreima 4741 . . . 4 (𝐹B) = {x A𝐶 B}
15 fimacnv 5221 . . . 4 (𝐹:AB → (𝐹B) = A)
1614, 15syl5reqr 2069 . . 3 (𝐹:ABA = {x A𝐶 B})
17 rabid2 2464 . . 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 1228   wcel 1374  {cab 2008  wral 2284  wrex 2285  {crab 2288  wss 2894  cmpt 3792  ccnv 4271  ran crn 4273  cima 4275   Fn wfn 4824  wf 4825
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fv 4837
This theorem is referenced by:  f1ompt  5245  fmpti  5246  fmptd  5247  rnmptss  5251  f1oresrab  5254  idref  5321  f1mpt  5335  f1stres  5709  f2ndres  5710  fmpt2x  5749  fmpt2co  5760  iunon  5821
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