Theorem ovelimab 7589

Description: Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Assertion

Ref	Expression
ovelimab	⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦)))

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

Proof of Theorem ovelimab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvelimab 6954	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑧 ∈ (𝐵 × 𝐶)(𝐹‘𝑧) = 𝐷))
2		fveq2 6882	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7414	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2822	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦))
5	4	eqeq1d 2771	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = 𝐷 ↔ (𝑥𝐹𝑦) = 𝐷))
6		eqcom 2776	. . . 4 ⊢ ((𝑥𝐹𝑦) = 𝐷 ↔ 𝐷 = (𝑥𝐹𝑦))
7	5, 6	bitrdi 290	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = 𝐷 ↔ 𝐷 = (𝑥𝐹𝑦)))
8	7	rexxp 5829	. 2 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)(𝐹‘𝑧) = 𝐷 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦))
9	1, 8	bitrdi 290	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ⊆ wss 3911  ⟨cop 4598   × cxp 5660   “ cima 5665   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7414

This theorem is referenced by:  imaeqexov  7649  imaeqalov  7650  dfz2  12610  elq  12974  elzs  28543  shsel  31607  ofrn2  32926  eulerpartlemgh  34713  aks6d1c2lem4  42819