Theorem ovelrn 5588

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 5585	. . 3 ⊢ (𝐹 Fn (A × B) → ran 𝐹 = {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)})
2	1	eleq2d 2104	. 2 ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = (x𝐹y)}))
3		elex 2560	. . . 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 5478	. . . . . 6 ⊢ ((𝐹 Fn (A × B) ∧ x ∈ A ∧ y ∈ B) → (x𝐹y) ∈ V)
6		eleq1 2097	. . . . . 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 1104	. . . 4 ⊢ ((𝐹 Fn (A × B) ∧ (x ∈ A ∧ y ∈ B)) → (𝐶 = (x𝐹y) → 𝐶 ∈ V))
9	8	rexlimdvva 2434	. . 3 ⊢ (𝐹 Fn (A × B) → (∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y) → 𝐶 ∈ V))
10		eqeq1 2043	. . . . . 6 ⊢ (z = 𝐶 → (z = (x𝐹y) ↔ 𝐶 = (x𝐹y)))
11	10	2rexbidv 2343	. . . . 5 ⊢ (z = 𝐶 → (∃x ∈ A ∃y ∈ B z = (x𝐹y) ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y)))
12	11	elabg 2682	. . . 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 620	. 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 884   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551   × cxp 4285  ran crn 4288   Fn wfn 4839  (class class class)co 5452

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fn 4847  df-fv 4852  df-ov 5455

This theorem is referenced by: (None)