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Theorem mpt2mptx2 41422
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 mpt2mptx 6711. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
mpt2mptx2.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2mptx2 (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 mpt2mptx2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4680 . 2 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)}
2 df-mpt2 6615 . . 3 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
3 eliunxp2 41421 . . . . . . 7 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
43anbi1i 730 . . . . . 6 ((𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
5 19.41vv 1912 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
6 anass 680 . . . . . . . 8 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
7 mpt2mptx2.1 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
87eqeq2d 2631 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶𝑤 = 𝐷))
98anbi2d 739 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
109pm5.32i 668 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
116, 10bitri 264 . . . . . . 7 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
12112exbii 1772 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
134, 5, 123bitr2i 288 . . . . 5 ((𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
1413opabbii 4684 . . . 4 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
15 dfoprab2 6661 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
1614, 15eqtr4i 2646 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
172, 16eqtr4i 2646 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)}
181, 17eqtr4i 2646 1 (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  {csn 4153  cop 4159   ciun 4490  {copab 4677  cmpt 4678   × cxp 5077  {coprab 6611  cmpt2 6612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-iun 4492  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-oprab 6614  df-mpt2 6615
This theorem is referenced by:  dmmpt2ssx2  41424
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