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Theorem mptpreima 4814
 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 3820 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2060 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43cnveqi 4510 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
5 cnvopab 4726 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
64, 5eqtri 2060 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76imaeq1i 4665 . 2 (𝐹𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶)
8 df-ima 4358 . . 3 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶)
9 resopab 4652 . . . . 5 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
109rneqi 4562 . . . 4 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
11 ancom 253 . . . . . . . . 9 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶))
12 anass 381 . . . . . . . . 9 (((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1311, 12bitri 173 . . . . . . . 8 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1413exbii 1496 . . . . . . 7 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
15 19.42v 1786 . . . . . . . 8 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)))
16 df-clel 2036 . . . . . . . . . 10 (𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦𝐶))
1716bicomi 123 . . . . . . . . 9 (∃𝑦(𝑦 = 𝐵𝑦𝐶) ↔ 𝐵𝐶)
1817anbi2i 430 . . . . . . . 8 ((𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
1915, 18bitri 173 . . . . . . 7 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
2014, 19bitri 173 . . . . . 6 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴𝐵𝐶))
2120abbii 2153 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
22 rnopab 4581 . . . . 5 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
23 df-rab 2315 . . . . 5 {𝑥𝐴𝐵𝐶} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
2421, 22, 233eqtr4i 2070 . . . 4 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥𝐴𝐵𝐶}
2510, 24eqtri 2060 . . 3 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {𝑥𝐴𝐵𝐶}
268, 25eqtri 2060 . 2 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = {𝑥𝐴𝐵𝐶}
277, 26eqtri 2060 1 (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  {crab 2310  {copab 3817   ↦ cmpt 3818  ◡ccnv 4344  ran crn 4346   ↾ cres 4347   “ cima 4348 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  mptiniseg  4815  dmmpt  4816  fmpt  5319  f1oresrab  5329  suppssfv  5708  suppssov1  5709
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