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Theorem mptpreima 4841
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
mptpreima (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
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 = (𝑥𝐴𝐵)
2 df-mpt 3847 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2076 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43cnveqi 4537 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
5 cnvopab 4753 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
64, 5eqtri 2076 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76imaeq1i 4692 . 2 (𝐹𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶)
8 df-ima 4385 . . 3 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶)
9 resopab 4679 . . . . 5 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
109rneqi 4589 . . . 4 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
11 ancom 257 . . . . . . . . 9 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶))
12 anass 387 . . . . . . . . 9 (((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1311, 12bitri 177 . . . . . . . 8 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1413exbii 1512 . . . . . . 7 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
15 19.42v 1802 . . . . . . . 8 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)))
16 df-clel 2052 . . . . . . . . . 10 (𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦𝐶))
1716bicomi 127 . . . . . . . . 9 (∃𝑦(𝑦 = 𝐵𝑦𝐶) ↔ 𝐵𝐶)
1817anbi2i 438 . . . . . . . 8 ((𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
1915, 18bitri 177 . . . . . . 7 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
2014, 19bitri 177 . . . . . 6 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴𝐵𝐶))
2120abbii 2169 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
22 rnopab 4608 . . . . 5 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
23 df-rab 2332 . . . . 5 {𝑥𝐴𝐵𝐶} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
2421, 22, 233eqtr4i 2086 . . . 4 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥𝐴𝐵𝐶}
2510, 24eqtri 2076 . . 3 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {𝑥𝐴𝐵𝐶}
268, 25eqtri 2076 . 2 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = {𝑥𝐴𝐵𝐶}
277, 26eqtri 2076 1 (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  {crab 2327  {copab 3844  cmpt 3845  ccnv 4371  ran crn 4373  cres 4374  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-mpt 3847  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  mptiniseg  4842  dmmpt  4843  fmpt  5346  f1oresrab  5356  suppssfv  5735  suppssov1  5736
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