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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 1803 | . 2 ⊢ ∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) |
3 | fnoprabg 7389 | . 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 396 ∀wal 1540 ∃!weu 2570 {copab 5141 Fn wfn 6426 {coprab 7270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-fun 6433 df-fn 6434 df-oprab 7273 |
This theorem is referenced by: ovid 7406 ov 7409 |
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