Theorem mpomptx 5862

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.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑥,𝐶,𝑦   𝑧,𝐷

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem mpomptx

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 3991	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
2		df-mpo 5779	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)}
3		eliunxp 4678	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4	3	anbi1i 453	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶))
5		19.41vv 1875	. . . . . 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 398	. . . . . . . 8 ⊢ (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)))
7		mpompt.1	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
8	7	eqeq2d 2151	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶 ↔ 𝑤 = 𝐷))
9	8	anbi2d 459	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
10	9	pm5.32i 449	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
11	6, 10	bitri 183	. . . . . . 7 ⊢ (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
12	11	2exbii 1585	. . . . . 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 207	. . . . 5 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)))
14	13	opabbii 3995	. . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷))}
15		dfoprab2 5818	. . . 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 2163	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐷)}
17	2, 16	eqtr4i 2163	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
18	1, 17	eqtr4i 2163	1 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {csn 3527  ⟨cop 3530  ∪ ciun 3813  {copab 3988   ↦ cmpt 3989   × cxp 4537  {coprab 5775   ∈ cmpo 5776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  mpompt  5863  mpomptsx  6095  dmmpossx  6097  fmpox  6098