Theorem ffnov 6806

Description: An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)

Assertion

Ref	Expression
ffnov	⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶))

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

Proof of Theorem ffnov

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffnfv 6428	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶))
2		fveq2 6229	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 6693	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	syl6eqr 2703	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eleq1d 2715	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶))
6	5	ralxp 5296	. . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)
7	6	anbi2i 730	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶))
8	1, 7	bitri 264	1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ⟨cop 4216   × cxp 5141   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693

This theorem is referenced by:  fovcl  6807  cantnfvalf  8600  axaddf  10004  axmulf  10005  mulnzcnopr  10711  frmdplusg  17438  gass  17780  sylow2blem2  18082  matecl  20279  txdis1cn  21486  isxmet2d  22179  prdsmet  22222  imasdsf1olem  22225  imasf1oxmet  22227  imasf1omet  22228  xmetresbl  22289  comet  22365  tgqioo  22650  xrtgioo  22656  opnmblALT  23417  dvdsmulf1o  24965  hhssabloilem  28246  fovcld  29568  pstmxmet  30068  xrge0pluscn  30114  isbndx  33711  isbnd3  33713  isbnd3b  33714  prdsbnd  33722  isdrngo2  33887  clintopcllaw  42172