Theorem brfns 5833

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 4689	. . 3 ⊢ (F Fns A → (F ∈ V ∧ A ∈ V))
2	1	simprd 449	. 2 ⊢ (F Fns A → A ∈ V)
3		fndm 5182	. . . 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 5105	. . . 4 ⊢ (F ∈ V → dom F ∈ V)
7	5, 6	ax-mp 8	. . 3 ⊢ dom F ∈ V
8	4, 7	syl6eqel 2441	. 2 ⊢ (F Fn A → A ∈ V)
9		breq2 4643	. . 3 ⊢ (a = A → (F Fns a ↔ F Fns A))
10		fneq2 5174	. . 3 ⊢ (a = A → (F Fn a ↔ F Fn A))
11		vex 2862	. . . 4 ⊢ a ∈ V
12		fneq1 5173	. . . 4 ⊢ (f = F → (f Fn b ↔ F Fn b))
13		fneq2 5174	. . . 4 ⊢ (b = a → (F Fn b ↔ F Fn a))
14		df-fns 5762	. . . 4 ⊢ Fns = {⟨f, b⟩ ∣ f Fn b}
15	5, 11, 12, 13, 14	brab 4709	. . 3 ⊢ (F Fns a ↔ F Fn a)
16	9, 10, 15	vtoclbg 2915	. 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 2859   class class class wbr 4639  dom cdm 4772   Fn wfn 4776   Fns cfns 5761

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-swap 4724  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fns 5762

This theorem is referenced by:  enex  6031  ovcelem1  6171  ceex  6174