Theorem fnov 5591

Description: Representation of an operation class abstraction in terms of its values. (Contributed by set.mm contributors, 7-Feb-2004.)

Assertion

Ref	Expression
fnov	⊢ (F Fn (A × B) ↔ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})

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

Proof of Theorem fnov

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 5363	. 2 ⊢ (F Fn (A × B) ↔ F = {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ z = (F ‘w))})
2		elxp 4801	. . . . . . 7 ⊢ (w ∈ (A × B) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
3	2	anbi1i 676	. . . . . 6 ⊢ ((w ∈ (A × B) ∧ z = (F ‘w)) ↔ (∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)))
4		19.41vv 1902	. . . . . 6 ⊢ (∃x∃y((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ (∃x∃y(w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)))
5		anass 630	. . . . . . . 8 ⊢ (((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (F ‘w))))
6		fveq2 5328	. . . . . . . . . . . 12 ⊢ (w = ⟨x, y⟩ → (F ‘w) = (F ‘⟨x, y⟩))
7		df-ov 5526	. . . . . . . . . . . 12 ⊢ (xFy) = (F ‘⟨x, y⟩)
8	6, 7	syl6eqr 2403	. . . . . . . . . . 11 ⊢ (w = ⟨x, y⟩ → (F ‘w) = (xFy))
9	8	eqeq2d 2364	. . . . . . . . . 10 ⊢ (w = ⟨x, y⟩ → (z = (F ‘w) ↔ z = (xFy)))
10	9	anbi2d 684	. . . . . . . . 9 ⊢ (w = ⟨x, y⟩ → (((x ∈ A ∧ y ∈ B) ∧ z = (F ‘w)) ↔ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
11	10	pm5.32i 618	. . . . . . . 8 ⊢ ((w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (F ‘w))) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
12	5, 11	bitri 240	. . . . . . 7 ⊢ (((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
13	12	2exbii 1583	. . . . . 6 ⊢ (∃x∃y((w = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)) ∧ z = (F ‘w)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
14	3, 4, 13	3bitr2i 264	. . . . 5 ⊢ ((w ∈ (A × B) ∧ z = (F ‘w)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))))
15	14	opabbii 4626	. . . 4 ⊢ {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ z = (F ‘w))} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy)))}
16		dfoprab2 5558	. . . 4 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy)))}
17	15, 16	eqtr4i 2376	. . 3 ⊢ {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ z = (F ‘w))} = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))}
18	17	eqeq2i 2363	. 2 ⊢ (F = {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ z = (F ‘w))} ↔ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})
19	1, 18	bitri 240	1 ⊢ (F Fn (A × B) ↔ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = (xFy))})

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622   × cxp 4770   Fn wfn 4776   ‘cfv 4781  (class class class)co 5525  {coprab 5527

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-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-ov 5526  df-oprab 5528

This theorem is referenced by:  fov  5592  fnov2  5707