Theorem iunfopab 5205

Description: Two ways to express a function as a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Sep-2011.) (Contributed by set.mm contributors, 19-Dec-2008.)

Hypothesis

Ref	Expression
iunfopab.1	⊢ B ∈ V

Assertion

Ref	Expression
iunfopab	⊢ ∪x ∈ A {⟨x, B⟩} = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}

Distinct variable groups:   y,A   y,B   x,y

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem iunfopab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2621	. . . 4 ⊢ (∃x ∈ A z ∈ {⟨x, B⟩} ↔ ∃x(x ∈ A ∧ z ∈ {⟨x, B⟩}))
2		vex 2863	. . . . . . . 8 ⊢ z ∈ V
3	2	elsnc 3757	. . . . . . 7 ⊢ (z ∈ {⟨x, B⟩} ↔ z = ⟨x, B⟩)
4	3	anbi2i 675	. . . . . 6 ⊢ ((x ∈ A ∧ z ∈ {⟨x, B⟩}) ↔ (x ∈ A ∧ z = ⟨x, B⟩))
5		iunfopab.1	. . . . . . 7 ⊢ B ∈ V
6		opeq2 4580	. . . . . . . . 9 ⊢ (y = B → ⟨x, y⟩ = ⟨x, B⟩)
7	6	eqeq2d 2364	. . . . . . . 8 ⊢ (y = B → (z = ⟨x, y⟩ ↔ z = ⟨x, B⟩))
8	7	anbi2d 684	. . . . . . 7 ⊢ (y = B → ((x ∈ A ∧ z = ⟨x, y⟩) ↔ (x ∈ A ∧ z = ⟨x, B⟩)))
9	5, 8	ceqsexv 2895	. . . . . 6 ⊢ (∃y(y = B ∧ (x ∈ A ∧ z = ⟨x, y⟩)) ↔ (x ∈ A ∧ z = ⟨x, B⟩))
10		an13 774	. . . . . . 7 ⊢ ((y = B ∧ (x ∈ A ∧ z = ⟨x, y⟩)) ↔ (z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B)))
11	10	exbii 1582	. . . . . 6 ⊢ (∃y(y = B ∧ (x ∈ A ∧ z = ⟨x, y⟩)) ↔ ∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B)))
12	4, 9, 11	3bitr2i 264	. . . . 5 ⊢ ((x ∈ A ∧ z ∈ {⟨x, B⟩}) ↔ ∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B)))
13	12	exbii 1582	. . . 4 ⊢ (∃x(x ∈ A ∧ z ∈ {⟨x, B⟩}) ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B)))
14	1, 13	bitri 240	. . 3 ⊢ (∃x ∈ A z ∈ {⟨x, B⟩} ↔ ∃x∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B)))
15	14	abbii 2466	. 2 ⊢ {z ∣ ∃x ∈ A z ∈ {⟨x, B⟩}} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B))}
16		df-iun 3972	. 2 ⊢ ∪x ∈ A {⟨x, B⟩} = {z ∣ ∃x ∈ A z ∈ {⟨x, B⟩}}
17		df-opab 4624	. 2 ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)} = {z ∣ ∃x∃y(z = ⟨x, y⟩ ∧ (x ∈ A ∧ y = B))}
18	15, 16, 17	3eqtr4i 2383	1 ⊢ ∪x ∈ A {⟨x, B⟩} = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860  {csn 3738  ∪ciun 3970  ⟨cop 4562  {copab 4623

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624

This theorem is referenced by: (None)