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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 2523 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 |
3 | fnopab.2 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | fnopabg 5321 | . 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 1348 ∃!weu 2019 ∈ wcel 2141 ∀wral 2448 {copab 4049 Fn wfn 5193 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-fun 5200 df-fn 5201 |
This theorem is referenced by: fvopab3g 5569 |
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