Theorem ovelrn 5541

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 5538	. . 3 ⊢ (𝐹 Fn (A × B) → ran 𝐹 = {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)})
2	1	eleq2d 2089	. 2 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)}))
3		elex 2541	. . . 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 5431	. . . . . 6 ⊢ ((𝐹 Fn (A × B) ∧ x ∈ A ∧ y ∈ B) → (x𝐹y) ∈ V)
6		eleq1 2082	. . . . . 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 1093	. . . 4 ⊢ ((𝐹 Fn (A × B) ∧ (x ∈ A ∧ y ∈ B)) → (𝐶 = (x𝐹y) → 𝐶 ∈ V))
9	8	rexlimdvva 2416	. . 3 ⊢ (𝐹 Fn (A × B) → (∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y) → 𝐶 ∈ V))
10		eqeq1 2028	. . . . . 6 ⊢ (z = 𝐶 → (z = (x𝐹y) ↔ 𝐶 = (x𝐹y)))
11	10	2rexbidv 2325	. . . . 5 ⊢ (z = 𝐶 → (∃x ∈ A ∃y ∈ B z = (x𝐹y) ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))
12	11	elabg 2663	. . . 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 875   = wceq 1373   ∈ wcel 1375  {cab 2008  ∃wrex 2283  Vcvv 2533   × cxp 4236  ran crn 4239   Fn wfn 4791  (class class class)co 5405

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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-csb 2829  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804  df-ov 5408

This theorem is referenced by: (None)