Theorem mpt2mptx 5709

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
mpt2mpt.1	⊢ (z = ⟨x, y⟩ → C = D)

Assertion

Ref	Expression
mpt2mptx	⊢ (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 mpt2mptx

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 5653	. 2 ⊢ (z ∈ ∪x ∈ A ({x} × B) ↦ C) = {⟨z, w⟩ ∣ (z ∈ ∪x ∈ A ({x} × B) ∧ w = C)}
2		df-mpt2 5655	. . 3 ⊢ (x ∈ A, y ∈ B ↦ D) = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ w = D)}
3		eliunxp 4822	. . . . . . 7 ⊢ (z ∈ ∪x ∈ A ({x} × B) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
4	3	anbi1i 676	. . . . . 6 ⊢ ((z ∈ ∪x ∈ A ({x} × B) ∧ 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 630	. . . . . . . 8 ⊢ (((z = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ w = C) ↔ (z = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ w = C)))
7		mpt2mpt.1	. . . . . . . . . . 11 ⊢ (z = ⟨x, y⟩ → C = D)
8	7	eqeq2d 2364	. . . . . . . . . 10 ⊢ (z = ⟨x, y⟩ → (w = C ↔ w = D))
9	8	anbi2d 684	. . . . . . . . 9 ⊢ (z = ⟨x, y⟩ → (((x ∈ A ∧ y ∈ B) ∧ w = C) ↔ ((x ∈ A ∧ y ∈ B) ∧ w = D)))
10	9	pm5.32i 618	. . . . . . . 8 ⊢ ((z = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ w = C)) ↔ (z = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ w = D)))
11	6, 10	bitri 240	. . . . . . 7 ⊢ (((z = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ w = C) ↔ (z = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ w = D)))
12	11	2exbii 1583	. . . . . 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 264	. . . . 5 ⊢ ((z ∈ ∪x ∈ A ({x} × B) ∧ w = C) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ w = D)))
14	13	opabbii 4627	. . . 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 5559	. . . 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 2376	. . 3 ⊢ {⟨z, w⟩ ∣ (z ∈ ∪x ∈ A ({x} × B) ∧ w = C)} = {⟨⟨x, y⟩, w⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ w = D)}
17	2, 16	eqtr4i 2376	. 2 ⊢ (x ∈ A, y ∈ B ↦ D) = {⟨z, w⟩ ∣ (z ∈ ∪x ∈ A ({x} × B) ∧ w = C)}
18	1, 17	eqtr4i 2376	1 ⊢ (z ∈ ∪x ∈ A ({x} × B) ↦ C) = (x ∈ A, y ∈ B ↦ D)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3738  ∪ciun 3970  ⟨cop 4562  {copab 4623   × cxp 4771  {coprab 5528   ↦ cmpt 5652   ↦ cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-xp 4785  df-oprab 5529  df-mpt 5653  df-mpt2 5655

This theorem is referenced by:  mpt2mpt  5710  fmpt2x  5731