Theorem pmvalg 6011

Description: The value of the partial mapping operation. (A ↑_pm B) is the set of all partial functions that map from B to A. (Contributed by set.mm contributors, 15-Nov-2007.) (Revised by set.mm contributors, 8-Sep-2013.)

Assertion

Ref	Expression
pmvalg	⊢ ((A ∈ C ∧ B ∈ D) → (A ↑pm B) = {f ∈ ℘(B × A) ∣ Fun f})

Distinct variable groups:   A,f   B,f

Allowed substitution hints:   C(f)   D(f)

Proof of Theorem pmvalg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ C → A ∈ V)
2		elex 2868	. 2 ⊢ (B ∈ D → B ∈ V)
3		df-rab 2624	. . . . 5 ⊢ {f ∈ ℘(B × A) ∣ Fun f} = {f ∣ (f ∈ ℘(B × A) ∧ Fun f)}
4		ancom 437	. . . . . . 7 ⊢ ((f ∈ ℘(B × A) ∧ Fun f) ↔ (Fun f ∧ f ∈ ℘(B × A)))
5		df-pw 3725	. . . . . . . . 9 ⊢ ℘(B × A) = {f ∣ f ⊆ (B × A)}
6	5	abeq2i 2461	. . . . . . . 8 ⊢ (f ∈ ℘(B × A) ↔ f ⊆ (B × A))
7	6	anbi2i 675	. . . . . . 7 ⊢ ((Fun f ∧ f ∈ ℘(B × A)) ↔ (Fun f ∧ f ⊆ (B × A)))
8	4, 7	bitri 240	. . . . . 6 ⊢ ((f ∈ ℘(B × A) ∧ Fun f) ↔ (Fun f ∧ f ⊆ (B × A)))
9	8	abbii 2466	. . . . 5 ⊢ {f ∣ (f ∈ ℘(B × A) ∧ Fun f)} = {f ∣ (Fun f ∧ f ⊆ (B × A))}
10	3, 9	eqtri 2373	. . . 4 ⊢ {f ∈ ℘(B × A) ∣ Fun f} = {f ∣ (Fun f ∧ f ⊆ (B × A))}
11		pmex 6006	. . . . 5 ⊢ ((B ∈ V ∧ A ∈ V) → {f ∣ (Fun f ∧ f ⊆ (B × A))} ∈ V)
12	11	ancoms 439	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {f ∣ (Fun f ∧ f ⊆ (B × A))} ∈ V)
13	10, 12	syl5eqel 2437	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → {f ∈ ℘(B × A) ∣ Fun f} ∈ V)
14		xpeq2 4800	. . . . . 6 ⊢ (x = A → (y × x) = (y × A))
15	14	pweqd 3728	. . . . 5 ⊢ (x = A → ℘(y × x) = ℘(y × A))
16		biidd 228	. . . . 5 ⊢ (x = A → (Fun f ↔ Fun f))
17	15, 16	rabeqbidv 2855	. . . 4 ⊢ (x = A → {f ∈ ℘(y × x) ∣ Fun f} = {f ∈ ℘(y × A) ∣ Fun f})
18		xpeq1 4799	. . . . . 6 ⊢ (y = B → (y × A) = (B × A))
19	18	pweqd 3728	. . . . 5 ⊢ (y = B → ℘(y × A) = ℘(B × A))
20		biidd 228	. . . . 5 ⊢ (y = B → (Fun f ↔ Fun f))
21	19, 20	rabeqbidv 2855	. . . 4 ⊢ (y = B → {f ∈ ℘(y × A) ∣ Fun f} = {f ∈ ℘(B × A) ∣ Fun f})
22		df-pm 6003	. . . 4 ⊢ ↑pm = (x ∈ V, y ∈ V ↦ {f ∈ ℘(y × x) ∣ Fun f})
23	17, 21, 22	ovmpt2g 5716	. . 3 ⊢ ((A ∈ V ∧ B ∈ V ∧ {f ∈ ℘(B × A) ∣ Fun f} ∈ V) → (A ↑pm B) = {f ∈ ℘(B × A) ∣ Fun f})
24	13, 23	mpd3an3 1278	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ↑pm B) = {f ∈ ℘(B × A) ∣ Fun f})
25	1, 2, 24	syl2an 463	1 ⊢ ((A ∈ C ∧ B ∈ D) → (A ↑pm B) = {f ∈ ℘(B × A) ∣ Fun f})

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  Vcvv 2860   ⊆ wss 3258  ℘cpw 3723   × cxp 4771  Fun wfun 4776  (class class class)co 5526   ↑_pm cpm 6001

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-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-funs 5761  df-pm 6003

This theorem is referenced by:  elpmg  6014  mapsspm  6022