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Theorem mptpreima 4958
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 3923 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2115 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43cnveqi 4642 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
5 cnvopab 4866 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
64, 5eqtri 2115 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76imaeq1i 4804 . 2 (𝐹𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶)
8 df-ima 4480 . . 3 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶)
9 resopab 4789 . . . . 5 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
109rneqi 4695 . . . 4 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
11 ancom 263 . . . . . . . . 9 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶))
12 anass 394 . . . . . . . . 9 (((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1311, 12bitri 183 . . . . . . . 8 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1413exbii 1548 . . . . . . 7 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
15 19.42v 1841 . . . . . . . 8 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)))
16 df-clel 2091 . . . . . . . . . 10 (𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦𝐶))
1716bicomi 131 . . . . . . . . 9 (∃𝑦(𝑦 = 𝐵𝑦𝐶) ↔ 𝐵𝐶)
1817anbi2i 446 . . . . . . . 8 ((𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
1915, 18bitri 183 . . . . . . 7 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
2014, 19bitri 183 . . . . . 6 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴𝐵𝐶))
2120abbii 2210 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
22 rnopab 4714 . . . . 5 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
23 df-rab 2379 . . . . 5 {𝑥𝐴𝐵𝐶} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
2421, 22, 233eqtr4i 2125 . . . 4 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥𝐴𝐵𝐶}
2510, 24eqtri 2115 . . 3 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {𝑥𝐴𝐵𝐶}
268, 25eqtri 2115 . 2 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = {𝑥𝐴𝐵𝐶}
277, 26eqtri 2115 1 (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1296  wex 1433  wcel 1445  {cab 2081  {crab 2374  {copab 3920  cmpt 3921  ccnv 4466  ran crn 4468  cres 4469  cima 4470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-mpt 3923  df-xp 4473  df-rel 4474  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480
This theorem is referenced by:  mptiniseg  4959  dmmpt  4960  fmpt  5488  f1oresrab  5502  suppssfv  5890  suppssov1  5891  infrenegsupex  9181  infxrnegsupex  10806
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