Theorem dffn5 5364

Description: Representation of a function in terms of its values. (Contributed by set.mm contributors, 29-Jan-2004.)

Assertion

Ref	Expression
dffn5	⊢ (F Fn A ↔ F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))})

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem dffn5

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnop 5187	. . . . . . 7 ⊢ ((F Fn A ∧ ⟨z, w⟩ ∈ F) → z ∈ A)
2	1	ex 423	. . . . . 6 ⊢ (F Fn A → (⟨z, w⟩ ∈ F → z ∈ A))
3	2	pm4.71rd 616	. . . . 5 ⊢ (F Fn A → (⟨z, w⟩ ∈ F ↔ (z ∈ A ∧ ⟨z, w⟩ ∈ F)))
4		eqcom 2355	. . . . . . 7 ⊢ (w = (F ‘z) ↔ (F ‘z) = w)
5		fnopfvb 5360	. . . . . . 7 ⊢ ((F Fn A ∧ z ∈ A) → ((F ‘z) = w ↔ ⟨z, w⟩ ∈ F))
6	4, 5	syl5bb 248	. . . . . 6 ⊢ ((F Fn A ∧ z ∈ A) → (w = (F ‘z) ↔ ⟨z, w⟩ ∈ F))
7	6	pm5.32da 622	. . . . 5 ⊢ (F Fn A → ((z ∈ A ∧ w = (F ‘z)) ↔ (z ∈ A ∧ ⟨z, w⟩ ∈ F)))
8	3, 7	bitr4d 247	. . . 4 ⊢ (F Fn A → (⟨z, w⟩ ∈ F ↔ (z ∈ A ∧ w = (F ‘z))))
9		vex 2863	. . . . 5 ⊢ z ∈ V
10		vex 2863	. . . . 5 ⊢ w ∈ V
11		eleq1 2413	. . . . . 6 ⊢ (x = z → (x ∈ A ↔ z ∈ A))
12		fveq2 5329	. . . . . . 7 ⊢ (x = z → (F ‘x) = (F ‘z))
13	12	eqeq2d 2364	. . . . . 6 ⊢ (x = z → (y = (F ‘x) ↔ y = (F ‘z)))
14	11, 13	anbi12d 691	. . . . 5 ⊢ (x = z → ((x ∈ A ∧ y = (F ‘x)) ↔ (z ∈ A ∧ y = (F ‘z))))
15		eqeq1 2359	. . . . . 6 ⊢ (y = w → (y = (F ‘z) ↔ w = (F ‘z)))
16	15	anbi2d 684	. . . . 5 ⊢ (y = w → ((z ∈ A ∧ y = (F ‘z)) ↔ (z ∈ A ∧ w = (F ‘z))))
17	9, 10, 14, 16	opelopab 4709	. . . 4 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))} ↔ (z ∈ A ∧ w = (F ‘z)))
18	8, 17	syl6bbr 254	. . 3 ⊢ (F Fn A → (⟨z, w⟩ ∈ F ↔ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))}))
19	18	eqrelrdv 4853	. 2 ⊢ (F Fn A → F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))})
20		fvex 5340	. . . 4 ⊢ (F ‘x) ∈ V
21		eqid 2353	. . . 4 ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))} = {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))}
22	20, 21	fnopab2 5209	. . 3 ⊢ {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))} Fn A
23		fneq1 5174	. . 3 ⊢ (F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))} → (F Fn A ↔ {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))} Fn A))
24	22, 23	mpbiri 224	. 2 ⊢ (F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))} → F Fn A)
25	19, 24	impbii 180	1 ⊢ (F Fn A ↔ F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = (F ‘x))})

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ⟨cop 4562  {copab 4623   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-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  fopabfv  5431  fnov  5592  dffn5v  5707