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| Mirrors > Home > MPE Home > Th. List > fnopab | Structured version Visualization version 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 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 | 
| 3 | fnopab.2 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 4 | 3 | fnopabg 6705 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴) | 
| 5 | 2, 4 | mpbi 230 | 1 ⊢ 𝐹 Fn 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 ∀wral 3061 {copab 5205 Fn wfn 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 | 
| This theorem is referenced by: fvopab3g 7011 | 
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