Theorem pmvalg 6010

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 2867	. 2 ⊢ (A ∈ C → A ∈ V)
2		elex 2867	. 2 ⊢ (B ∈ D → B ∈ V)
3		df-rab 2623	. . . . 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 3724	. . . . . . . . 9 ⊢ ℘(B × A) = {f ∣ f ⊆ (B × A)}
6	5	abeq2i 2460	. . . . . . . 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 2465	. . . . 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 6005	. . . . 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 4799	. . . . . 6 ⊢ (x = A → (y × x) = (y × A))
15	14	pweqd 3727	. . . . 5 ⊢ (x = A → ℘(y × x) = ℘(y × A))
16		biidd 228	. . . . 5 ⊢ (x = A → (Fun f ↔ Fun f))
17	15, 16	rabeqbidv 2854	. . . 4 ⊢ (x = A → {f ∈ ℘(y × x) ∣ Fun f} = {f ∈ ℘(y × A) ∣ Fun f})
18		xpeq1 4798	. . . . . 6 ⊢ (y = B → (y × A) = (B × A))
19	18	pweqd 3727	. . . . 5 ⊢ (y = B → ℘(y × A) = ℘(B × A))
20		biidd 228	. . . . 5 ⊢ (y = B → (Fun f ↔ Fun f))
21	19, 20	rabeqbidv 2854	. . . 4 ⊢ (y = B → {f ∈ ℘(y × A) ∣ Fun f} = {f ∈ ℘(B × A) ∣ Fun f})
22		df-pm 6002	. . . 4 ⊢ ↑pm = (x ∈ V, y ∈ V ↦ {f ∈ ℘(y × x) ∣ Fun f})
23	17, 21, 22	ovmpt2g 5715	. . 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 2618  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722   × cxp 4770  Fun wfun 4775  (class class class)co 5525   ↑_pm cpm 6000

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-funs 5760  df-pm 6002

This theorem is referenced by:  elpmg  6013  mapsspm  6021