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Mirrors > Home > MPE Home > Th. List > fnoprab | Structured version Visualization version 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 1793 | . 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) |
3 | fnoprabg 7269 | . 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 setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 ∃!weu 2649 {copab 5121 Fn wfn 6345 {coprab 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-fun 6352 df-fn 6353 df-oprab 7154 |
This theorem is referenced by: ovid 7285 ov 7288 |
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