Theorem mpomptxf 31016

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	⊢ Ⅎ𝑥𝐶
mpomptxf.1	⊢ Ⅎ𝑦𝐶
mpomptxf.2	⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)

Assertion

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

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.

Step	Hyp	Ref	Expression
1		df-mpt 5158	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
2		df-mpo 7280	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)}
3		eliunxp 5746	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	anbi1i 624	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
5		mpomptxf.1	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐶
6	5	nfeq2 2924	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 = 𝐶
7	6	19.41 2228	. . . . . . . 8 ⊢ (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
8	7	exbii 1850	. . . . . . 7 ⊢ (∃𝑥∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
9		mpomptxf.0	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
10	9	nfeq2 2924	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 = 𝐶
11	10	19.41 2228	. . . . . . 7 ⊢ (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
12	8, 11	bitri 274	. . . . . 6 ⊢ (∃𝑥∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
13		anass 469	. . . . . . . 8 ⊢ (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
14		mpomptxf.2	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
15	14	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶 ↔ 𝑤 = 𝐷))
16	15	anbi2d 629	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
17	16	pm5.32i 575	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
18	13, 17	bitri 274	. . . . . . 7 ⊢ (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
19	18	2exbii 1851	. . . . . 6 ⊢ (∃𝑥∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
20	4, 12, 19	3bitr2i 299	. . . . 5 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
21	20	opabbii 5141	. . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷))}
22		dfoprab2 7333	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷))}
23	21, 22	eqtr4i 2769	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)}
24	2, 23	eqtr4i 2769	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
25	1, 24	eqtr4i 2769	1 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Ⅎwnfc 2887  {csn 4561  ⟨cop 4567  ∪ ciun 4924  {copab 5136   ↦ cmpt 5157   × cxp 5587  {coprab 7276   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-iun 4926  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-oprab 7279  df-mpo 7280

This theorem is referenced by:  gsummpt2co  31308