Theorem ovelrn 6001

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 5998	. . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})
2	1	eleq2d 2240	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}))
3		elex 2741	. . . 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 5886	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐹𝑦) ∈ V)
6		eleq1 2233	. . . . . 6 ⊢ (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V))
7	5, 6	syl5ibrcom 156	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
8	7	3expb 1199	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
9	8	rexlimdvva 2595	. . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V))
10		eqeq1 2177	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦)))
11	10	2rexbidv 2495	. . . . 5 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))
12	11	elabg 2876	. . . 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 700	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))
15	2, 14	bitrd 187	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {cab 2156  ∃wrex 2449  Vcvv 2730   × cxp 4609  ran crn 4612   Fn wfn 5193  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-ov 5856

This theorem is referenced by:  blrnps  13205  blrn  13206  tgioo  13340