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Theorem fnrnov 7576
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 ∣ ∃𝑥𝐴𝑦𝐵 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.
StepHypRef Expression
1 fnrnfv 6944 . 2 (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)})
2 fveq2 6884 . . . . . 6 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7407 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2784 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹𝑤) = (𝑥𝐹𝑦))
54eqeq2d 2737 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (𝐹𝑤) ↔ 𝑧 = (𝑥𝐹𝑦)))
65rexxp 5835 . . 3 (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦))
76abbii 2796 . 2 {𝑧 ∣ ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹𝑤)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)}
81, 7eqtrdi 2782 1 (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = (𝑥𝐹𝑦)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  {cab 2703  wrex 3064  cop 4629   × cxp 5667  ran crn 5670   Fn wfn 6531  cfv 6536  (class class class)co 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7407
This theorem is referenced by:  ovelrn  7579
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