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Mirrors > Home > ILE Home > Th. List > ovelrn | Structured version GIF version |
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.) |
Ref | Expression |
---|---|
ovelrn | ⊢ (𝐹 Fn (A × B) → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐶 = (x𝐹y))) |
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))) |
Colors of variables: wff set class |
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) |
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