Theorem brfns 5834

Description: Binary relationship form of Fns relationship. (Contributed by SF, 23-Feb-2015.)

Hypothesis

Ref	Expression
brfns.1	⊢ F ∈ V

Assertion

Ref	Expression
brfns	⊢ (F Fns A ↔ F Fn A)

Proof of Theorem brfns

Dummy variables a b f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 ⊢ (F Fns A → (F ∈ V ∧ A ∈ V))
2	1	simprd 449	. 2 ⊢ (F Fns A → A ∈ V)
3		fndm 5183	. . . 4 ⊢ (F Fn A → dom F = A)
4	3	eqcomd 2358	. . 3 ⊢ (F Fn A → A = dom F)
5		brfns.1	. . . 4 ⊢ F ∈ V
6		dmexg 5106	. . . 4 ⊢ (F ∈ V → dom F ∈ V)
7	5, 6	ax-mp 5	. . 3 ⊢ dom F ∈ V
8	4, 7	syl6eqel 2441	. 2 ⊢ (F Fn A → A ∈ V)
9		breq2 4644	. . 3 ⊢ (a = A → (F Fns a ↔ F Fns A))
10		fneq2 5175	. . 3 ⊢ (a = A → (F Fn a ↔ F Fn A))
11		vex 2863	. . . 4 ⊢ a ∈ V
12		fneq1 5174	. . . 4 ⊢ (f = F → (f Fn b ↔ F Fn b))
13		fneq2 5175	. . . 4 ⊢ (b = a → (F Fn b ↔ F Fn a))
14		df-fns 5763	. . . 4 ⊢ Fns = {⟨f, b⟩ ∣ f Fn b}
15	5, 11, 12, 13, 14	brab 4710	. . 3 ⊢ (F Fns a ↔ F Fn a)
16	9, 10, 15	vtoclbg 2916	. 2 ⊢ (A ∈ V → (F Fns A ↔ F Fn A))
17	2, 8, 16	pm5.21nii 342	1 ⊢ (F Fns A ↔ F Fn A)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640  dom cdm 4773   Fn wfn 4777   Fns cfns 5762

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-swap 4725  df-co 4727  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fns 5763

This theorem is referenced by:  enex  6032  ovcelem1  6172  ceex  6175