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Mirrors > Home > ILE Home > Th. List > fnrnov | GIF version |
Description: The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.) |
Ref | Expression |
---|---|
fnrnov | ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrnfv 5436 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)}) | |
2 | fveq2 5389 | . . . . . 6 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 5745 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | syl6eqr 2168 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝑥𝐹𝑦)) |
5 | 4 | eqeq2d 2129 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑧 = (𝐹‘𝑤) ↔ 𝑧 = (𝑥𝐹𝑦))) |
6 | 5 | rexxp 4653 | . . 3 ⊢ (∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)) |
7 | 6 | abbii 2233 | . 2 ⊢ {𝑧 ∣ ∃𝑤 ∈ (𝐴 × 𝐵)𝑧 = (𝐹‘𝑤)} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)} |
8 | 1, 7 | syl6eq 2166 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 {cab 2103 ∃wrex 2394 〈cop 3500 × cxp 4507 ran crn 4510 Fn wfn 5088 ‘cfv 5093 (class class class)co 5742 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-ov 5745 |
This theorem is referenced by: ovelrn 5887 |
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