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Theorem mptpreima 4757
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1 𝐹 = (x AB)
Assertion
Ref Expression
mptpreima (𝐹𝐶) = {x AB 𝐶}
Distinct variable group:   x,𝐶
Allowed substitution hints:   A(x)   B(x)   𝐹(x)

Proof of Theorem mptpreima
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dmmpt2.1 . . . . . 6 𝐹 = (x AB)
2 df-mpt 3811 . . . . . 6 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
31, 2eqtri 2057 . . . . 5 𝐹 = {⟨x, y⟩ ∣ (x A y = B)}
43cnveqi 4453 . . . 4 𝐹 = {⟨x, y⟩ ∣ (x A y = B)}
5 cnvopab 4669 . . . 4 {⟨x, y⟩ ∣ (x A y = B)} = {⟨y, x⟩ ∣ (x A y = B)}
64, 5eqtri 2057 . . 3 𝐹 = {⟨y, x⟩ ∣ (x A y = B)}
76imaeq1i 4608 . 2 (𝐹𝐶) = ({⟨y, x⟩ ∣ (x A y = B)} “ 𝐶)
8 df-ima 4301 . . 3 ({⟨y, x⟩ ∣ (x A y = B)} “ 𝐶) = ran ({⟨y, x⟩ ∣ (x A y = B)} ↾ 𝐶)
9 resopab 4595 . . . . 5 ({⟨y, x⟩ ∣ (x A y = B)} ↾ 𝐶) = {⟨y, x⟩ ∣ (y 𝐶 (x A y = B))}
109rneqi 4505 . . . 4 ran ({⟨y, x⟩ ∣ (x A y = B)} ↾ 𝐶) = ran {⟨y, x⟩ ∣ (y 𝐶 (x A y = B))}
11 ancom 253 . . . . . . . . 9 ((y 𝐶 (x A y = B)) ↔ ((x A y = B) y 𝐶))
12 anass 381 . . . . . . . . 9 (((x A y = B) y 𝐶) ↔ (x A (y = B y 𝐶)))
1311, 12bitri 173 . . . . . . . 8 ((y 𝐶 (x A y = B)) ↔ (x A (y = B y 𝐶)))
1413exbii 1493 . . . . . . 7 (y(y 𝐶 (x A y = B)) ↔ y(x A (y = B y 𝐶)))
15 19.42v 1783 . . . . . . . 8 (y(x A (y = B y 𝐶)) ↔ (x A y(y = B y 𝐶)))
16 df-clel 2033 . . . . . . . . . 10 (B 𝐶y(y = B y 𝐶))
1716bicomi 123 . . . . . . . . 9 (y(y = B y 𝐶) ↔ B 𝐶)
1817anbi2i 430 . . . . . . . 8 ((x A y(y = B y 𝐶)) ↔ (x A B 𝐶))
1915, 18bitri 173 . . . . . . 7 (y(x A (y = B y 𝐶)) ↔ (x A B 𝐶))
2014, 19bitri 173 . . . . . 6 (y(y 𝐶 (x A y = B)) ↔ (x A B 𝐶))
2120abbii 2150 . . . . 5 {xy(y 𝐶 (x A y = B))} = {x ∣ (x A B 𝐶)}
22 rnopab 4524 . . . . 5 ran {⟨y, x⟩ ∣ (y 𝐶 (x A y = B))} = {xy(y 𝐶 (x A y = B))}
23 df-rab 2309 . . . . 5 {x AB 𝐶} = {x ∣ (x A B 𝐶)}
2421, 22, 233eqtr4i 2067 . . . 4 ran {⟨y, x⟩ ∣ (y 𝐶 (x A y = B))} = {x AB 𝐶}
2510, 24eqtri 2057 . . 3 ran ({⟨y, x⟩ ∣ (x A y = B)} ↾ 𝐶) = {x AB 𝐶}
268, 25eqtri 2057 . 2 ({⟨y, x⟩ ∣ (x A y = B)} “ 𝐶) = {x AB 𝐶}
277, 26eqtri 2057 1 (𝐹𝐶) = {x AB 𝐶}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  {crab 2304  {copab 3808  cmpt 3809  ccnv 4287  ran crn 4289  cres 4290  cima 4291
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-bnd 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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  mptiniseg  4758  dmmpt  4759  fmpt  5262  f1oresrab  5272  suppssfv  5650  suppssov1  5651
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