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Theorem mpomptx2 44211
 Description: Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐴(𝑦) is not assumed to be constant w.r.t 𝑦, analogous to mpomptx 7258. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
mpomptx2.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpomptx2 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝐶,𝑦   𝑧,𝐷
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem mpomptx2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 5143 . 2 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)}
2 df-mpo 7156 . . 3 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
3 eliunxp2 44210 . . . . . . 7 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
43anbi1i 623 . . . . . 6 ((𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
5 19.41vv 1944 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
6 anass 469 . . . . . . . 8 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
7 mpomptx2.1 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
87eqeq2d 2836 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶𝑤 = 𝐷))
98anbi2d 628 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
109pm5.32i 575 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
116, 10bitri 276 . . . . . . 7 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
12112exbii 1842 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
134, 5, 123bitr2i 300 . . . . 5 ((𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
1413opabbii 5129 . . . 4 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
15 dfoprab2 7207 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
1614, 15eqtr4i 2851 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
172, 16eqtr4i 2851 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)}
181, 17eqtr4i 2851 1 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {csn 4563  ⟨cop 4569  ∪ ciun 4916  {copab 5124   ↦ cmpt 5142   × cxp 5551  {coprab 7152   ∈ cmpo 7153 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-iun 4918  df-opab 5125  df-mpt 5143  df-xp 5559  df-rel 5560  df-oprab 7155  df-mpo 7156 This theorem is referenced by:  dmmpossx2  44213
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