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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 1427 | . 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) |
3 | fnoprabg 5912 | . 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 1330 ∃!weu 2003 {copab 4020 Fn wfn 5158 {coprab 5815 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-rex 2438 df-v 2711 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-br 3962 df-opab 4022 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-fun 5165 df-fn 5166 df-oprab 5818 |
This theorem is referenced by: ovid 5927 ov 5930 |
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