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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 2540 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 |
3 | fnopab.2 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | fnopabg 5351 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴) |
5 | 2, 4 | mpbi 145 | 1 ⊢ 𝐹 Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∃!weu 2036 ∈ wcel 2158 ∀wral 2465 {copab 4075 Fn wfn 5223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-fun 5230 df-fn 5231 |
This theorem is referenced by: fvopab3g 5602 |
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