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Mirrors > Home > ILE Home > Th. List > fnoprab | GIF version |
Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
fnoprab.1 | ⊢ (𝜑 → ∃!𝑧𝜓) |
Ref | Expression |
---|---|
fnoprab | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝜑 ∧ 𝜓)} Fn {〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnoprab.1 | . . 3 ⊢ (𝜑 → ∃!𝑧𝜓) | |
2 | 1 | gen2 1430 | . 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) |
3 | fnoprabg 5923 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝜑 ∧ 𝜓)} Fn {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝜑 ∧ 𝜓)} Fn {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1333 ∃!weu 2006 {copab 4025 Fn wfn 5166 {coprab 5826 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-fun 5173 df-fn 5174 df-oprab 5829 |
This theorem is referenced by: ovid 5938 ov 5941 |
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