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Mirrors > Home > ILE Home > Th. List > ovelrn | 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 5588 | . . 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 5481 | . . . . . 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))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 {cab 2023 ∃wrex 2301 Vcvv 2551 × cxp 4286 ran crn 4289 Fn wfn 4840 (class class class)co 5455 |
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-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
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 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 df-ov 5458 |
This theorem is referenced by: (None) |
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