Theorem rnmpt2 5718

Description: The range of an operation given by the "maps to" notation. (Contributed by FL, 20-Jun-2011.)

Hypothesis

Ref	Expression
rngop.1	⊢ F = (x ∈ A, y ∈ B ↦ C)

Assertion

Ref	Expression
rnmpt2	⊢ ran F = {z ∣ ∃x ∈ A ∃y ∈ B z = C}

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

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

Proof of Theorem rnmpt2

Step	Hyp	Ref	Expression
1		rngop.1	. . . 4 ⊢ F = (x ∈ A, y ∈ B ↦ C)
2		df-mpt2 5655	. . . 4 ⊢ (x ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
3	1, 2	eqtri 2373	. . 3 ⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
4	3	rneqi 4958	. 2 ⊢ ran F = ran {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
5		rnoprab 5577	. 2 ⊢ ran {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)} = {z ∣ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ z = C)}
6		r2ex 2653	. . . 4 ⊢ (∃x ∈ A ∃y ∈ B z = C ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ z = C))
7	6	bicomi 193	. . 3 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ z = C) ↔ ∃x ∈ A ∃y ∈ B z = C)
8	7	abbii 2466	. 2 ⊢ {z ∣ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ z = C)} = {z ∣ ∃x ∈ A ∃y ∈ B z = C}
9	4, 5, 8	3eqtri 2377	1 ⊢ ran F = {z ∣ ∃x ∈ A ∃y ∈ B z = C}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ran crn 4774  {coprab 5528   ↦ 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-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-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-br 4641  df-ima 4728  df-rn 4787  df-oprab 5529  df-mpt2 5655

This theorem is referenced by: (None)