Theorem ffnov 7494

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 7073	. 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶))
2		fveq2 6842	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7371	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eleq1d 2822	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶))
6	5	ralxp 5798	. . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)
7	6	anbi2i 624	. 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 275	1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4588   × cxp 5630   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371

This theorem is referenced by:  fovcld  7495  cantnfvalf  9586  axaddf  11068  axmulf  11069  mulnzcnf  11795  frmdplusg  18791  gass  19245  sylow2blem2  19565  matecl  22384  txdis1cn  23594  isxmet2d  24286  prdsmet  24329  imasdsf1olem  24332  imasf1oxmet  24334  imasf1omet  24335  xmetresbl  24396  comet  24472  tgqioo  24759  xrtgioo  24766  opnmblALT  25575  mpodvdsmulf1o  27175  dvdsmulf1o  27177  hhssabloilem  31353  pstmxmet  34079  xrge0pluscn  34122  mpomulnzcnf  36519  isbndx  38037  isbnd3  38039  isbnd3b  38040  prdsbnd  38048  isdrngo2  38213  aks6d1c6lem3  42546  clintopcllaw  48575