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Theorem mpomptxf 31303
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.) (Revised by Thierry Arnoux, 31-Mar-2018.)
Hypotheses
Ref Expression
mpomptxf.0 𝑥𝐶
mpomptxf.1 𝑦𝐶
mpomptxf.2 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpomptxf (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem mpomptxf
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 5176 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
2 df-mpo 7342 . . 3 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
3 eliunxp 5779 . . . . . . 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 mpomptxf.1 . . . . . . . . . 10 𝑦𝐶
65nfeq2 2921 . . . . . . . . 9 𝑦 𝑤 = 𝐶
7619.41 2227 . . . . . . . 8 (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
87exbii 1849 . . . . . . 7 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
9 mpomptxf.0 . . . . . . . . 9 𝑥𝐶
109nfeq2 2921 . . . . . . . 8 𝑥 𝑤 = 𝐶
111019.41 2227 . . . . . . 7 (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
128, 11bitri 274 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
13 anass 469 . . . . . . . 8 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
14 mpomptxf.2 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
1514eqeq2d 2747 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶𝑤 = 𝐷))
1615anbi2d 629 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
1716pm5.32i 575 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
1813, 17bitri 274 . . . . . . 7 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
19182exbii 1850 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
204, 12, 193bitr2i 298 . . . . 5 ((𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
2120opabbii 5159 . . . 4 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
22 dfoprab2 7395 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
2321, 22eqtr4i 2767 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
242, 23eqtr4i 2767 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
251, 24eqtr4i 2767 1 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wex 1780  wcel 2105  wnfc 2884  {csn 4573  cop 4579   ciun 4941  {copab 5154  cmpt 5175   × cxp 5618  {coprab 7338  cmpo 7339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-iun 4943  df-opab 5155  df-mpt 5176  df-xp 5626  df-rel 5627  df-oprab 7341  df-mpo 7342
This theorem is referenced by:  gsummpt2co  31595
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