Theorem mpt2mptx2 41906

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 6649. (Contributed by AV, 30-Mar-2019.)

Hypothesis

Ref	Expression
mpt2mptx2.1	⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)

Assertion

Ref	Expression
mpt2mptx2	⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

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.

Step	Hyp	Ref	Expression
1		df-mpt 4645	. 2 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)}
2		df-mpt2 6554	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)}
3		eliunxp2 41905	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	anbi1i 727	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
5		19.41vv 1902	. . . . . 6 ⊢ (∃𝑥∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
6		anass 679	. . . . . . . 8 ⊢ (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
7		mpt2mptx2.1	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
8	7	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶 ↔ 𝑤 = 𝐷))
9	8	anbi2d 736	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
10	9	pm5.32i 667	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
11	6, 10	bitri 263	. . . . . . 7 ⊢ (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
12	11	2exbii 1765	. . . . . 6 ⊢ (∃𝑥∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
13	4, 5, 12	3bitr2i 287	. . . . 5 ⊢ ((𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
14	13	opabbii 4649	. . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷))}
15		dfoprab2 6599	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷))}
16	14, 15	eqtr4i 2635	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)}
17	2, 16	eqtr4i 2635	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∧ 𝑤 = 𝐶)}
18	1, 17	eqtr4i 2635	1 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {csn 4125  ⟨cop 4131  ∪ ciun 4455  {copab 4642   ↦ cmpt 4643   × cxp 5036  {coprab 6550   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-iun 4457  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  dmmpt2ssx2  41908