Theorem fnrnov 7585

Description: The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)

Assertion

Ref	Expression
fnrnov	⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {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 fnrnov

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnfv 6943	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)})
2		fveq2 6881	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7413	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦))
5	4	eqeq2d 2747	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹‘𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
6	5	rexxp 5827	. . 3 ⊢ (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))
7	6	abbii 2803	. 2 ⊢ {𝑧 ∣ ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}
8	1, 7	eqtrdi 2787	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})

Syntax hints:   → wi 4   = wceq 1540  {cab 2714  ∃wrex 3061  ⟨cop 4612   × cxp 5657  ran crn 5660   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7413

This theorem is referenced by:  ovelrn  7588