Theorem funfv 5376

Description: A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)

Assertion

Ref	Expression
funfv	⊢ (Fun F → (F ‘A) = ∪(F “ {A}))

Proof of Theorem funfv

Step	Hyp	Ref	Expression
1		fvex 5340	. . . . 5 ⊢ (F ‘A) ∈ V
2	1	unisn 3908	. . . 4 ⊢ ∪{(F ‘A)} = (F ‘A)
3		eqid 2353	. . . . . . 7 ⊢ dom F = dom F
4		df-fn 4791	. . . . . . 7 ⊢ (F Fn dom F ↔ (Fun F ∧ dom F = dom F))
5	3, 4	mpbiran2 885	. . . . . 6 ⊢ (F Fn dom F ↔ Fun F)
6		fnsnfv 5374	. . . . . 6 ⊢ ((F Fn dom F ∧ A ∈ dom F) → {(F ‘A)} = (F “ {A}))
7	5, 6	sylanbr 459	. . . . 5 ⊢ ((Fun F ∧ A ∈ dom F) → {(F ‘A)} = (F “ {A}))
8	7	unieqd 3903	. . . 4 ⊢ ((Fun F ∧ A ∈ dom F) → ∪{(F ‘A)} = ∪(F “ {A}))
9	2, 8	syl5eqr 2399	. . 3 ⊢ ((Fun F ∧ A ∈ dom F) → (F ‘A) = ∪(F “ {A}))
10	9	ex 423	. 2 ⊢ (Fun F → (A ∈ dom F → (F ‘A) = ∪(F “ {A})))
11		ndmfv 5350	. . 3 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅)
12		ndmima 5026	. . . . 5 ⊢ (¬ A ∈ dom F → (F “ {A}) = ∅)
13	12	unieqd 3903	. . . 4 ⊢ (¬ A ∈ dom F → ∪(F “ {A}) = ∪∅)
14		uni0 3919	. . . 4 ⊢ ∪∅ = ∅
15	13, 14	syl6eq 2401	. . 3 ⊢ (¬ A ∈ dom F → ∪(F “ {A}) = ∅)
16	11, 15	eqtr4d 2388	. 2 ⊢ (¬ A ∈ dom F → (F ‘A) = ∪(F “ {A}))
17	10, 16	pm2.61d1 151	1 ⊢ (Fun F → (F ‘A) = ∪(F “ {A}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∅c0 3551  {csn 3738  ∪cuni 3892   “ cima 4723  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   ‘cfv 4782

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-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  funfv2  5377  fvun  5379