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Mirrors > Home > ILE Home > Th. List > fnopab | GIF version |
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.) |
Ref | Expression |
---|---|
fnopab.1 | ⊢ (𝑥 ∈ 𝐴 → ∃!𝑦𝜑) |
fnopab.2 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Ref | Expression |
---|---|
fnopab | ⊢ 𝐹 Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnopab.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∃!𝑦𝜑) | |
2 | 1 | rgen 2444 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 |
3 | fnopab.2 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | fnopabg 5182 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴) |
5 | 2, 4 | mpbi 144 | 1 ⊢ 𝐹 Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∈ wcel 1448 ∃!weu 1960 ∀wral 2375 {copab 3928 Fn wfn 5054 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-fun 5061 df-fn 5062 |
This theorem is referenced by: fvopab3g 5426 |
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