Theorem ovelrn 5570

Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Assertion

Ref	Expression
ovelrn	⊢ (𝐹 Fn (A × B) → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))

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

Proof of Theorem ovelrn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnov 5567	. . 3 ⊢ (𝐹 Fn (A × B) → ran 𝐹 = {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)})
2	1	eleq2d 2090	. 2 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)}))
3		elex 2542	. . . 4 ⊢ (𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)} → 𝐶 ∈ V)
4	3	a1i 9	. . 3 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)} → 𝐶 ∈ V))
5		fnovex 5460	. . . . . 6 ⊢ ((𝐹 Fn (A × B) ∧ x ∈ A ∧ y ∈ B) → (x𝐹y) ∈ V)
6		eleq1 2083	. . . . . 6 ⊢ (𝐶 = (x𝐹y) → (𝐶 ∈ V ↔ (x𝐹y) ∈ V))
7	5, 6	syl5ibrcom 146	. . . . 5 ⊢ ((𝐹 Fn (A × B) ∧ x ∈ A ∧ y ∈ B) → (𝐶 = (x𝐹y) → 𝐶 ∈ V))
8	7	3expb 1091	. . . 4 ⊢ ((𝐹 Fn (A × B) ∧ (x ∈ A ∧ y ∈ B)) → (𝐶 = (x𝐹y) → 𝐶 ∈ V))
9	8	rexlimdvva 2417	. . 3 ⊢ (𝐹 Fn (A × B) → (∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y) → 𝐶 ∈ V))
10		eqeq1 2029	. . . . . 6 ⊢ (z = 𝐶 → (z = (x𝐹y) ↔ 𝐶 = (x𝐹y)))
11	10	2rexbidv 2326	. . . . 5 ⊢ (z = 𝐶 → (∃x ∈ A ∃y ∈ B z = (x𝐹y) ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))
12	11	elabg 2664	. . . 4 ⊢ (𝐶 ∈ V → (𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)} ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))
13	12	a1i 9	. . 3 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ V → (𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)} ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y))))
14	4, 9, 13	pm5.21ndd 608	. 2 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)} ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))
15	2, 14	bitrd 177	1 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 873   = wceq 1228   ∈ wcel 1375  {cab 2009  ∃wrex 2284  Vcvv 2534   × cxp 4268  ran crn 4271   Fn wfn 4822  (class class class)co 5434

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-csb 2829  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-iun 3632  df-br 3738  df-opab 3792  df-mpt 3793  df-id 4003  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-iota 4792  df-fun 4829  df-fn 4830  df-fv 4835  df-ov 5437

This theorem is referenced by: (None)