MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpomptx Structured version   Visualization version   GIF version

Theorem mpomptx 7546
Description: Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpompt.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpomptx (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑥,𝐶,𝑦   𝑧,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem mpomptx
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 5232 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
2 df-mpo 7436 . . 3 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
3 eliunxp 5851 . . . . . . 7 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
43anbi1i 624 . . . . . 6 ((𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
5 19.41vv 1948 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
6 anass 468 . . . . . . . 8 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
7 mpompt.1 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
87eqeq2d 2746 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶𝑤 = 𝐷))
98anbi2d 630 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
109pm5.32i 574 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
116, 10bitri 275 . . . . . . 7 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
12112exbii 1846 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
134, 5, 123bitr2i 299 . . . . 5 ((𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
1413opabbii 5215 . . . 4 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
15 dfoprab2 7491 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
1614, 15eqtr4i 2766 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
172, 16eqtr4i 2766 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
181, 17eqtr4i 2766 1 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {csn 4631  cop 4637   ciun 4996  {copab 5210  cmpt 5231   × cxp 5687  {coprab 7432  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  mpompt  7547  mpomptsx  8088  dmmpossx  8090  fmpox  8091  gsumcom2  20008
  Copyright terms: Public domain W3C validator