Theorem ovelimab 6046

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 5592	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑧 ∈ (𝐵 × 𝐶)(𝐹‘𝑧) = 𝐷))
2		fveq2 5534	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 5898	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2240	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦))
5	4	eqeq1d 2198	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = 𝐷 ↔ (𝑥𝐹𝑦) = 𝐷))
6		eqcom 2191	. . . 4 ⊢ ((𝑥𝐹𝑦) = 𝐷 ↔ 𝐷 = (𝑥𝐹𝑦))
7	5, 6	bitrdi 196	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = 𝐷 ↔ 𝐷 = (𝑥𝐹𝑦)))
8	7	rexxp 4789	. 2 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)(𝐹‘𝑧) = 𝐷 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦))
9	1, 8	bitrdi 196	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∃wrex 2469   ⊆ wss 3144  ⟨cop 3610   × cxp 4642   “ cima 4647   Fn wfn 5230  ‘cfv 5235  (class class class)co 5895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5898

This theorem is referenced by:  dfz2  9354  elq  9651