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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 3061 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 |
3 | fnopab.2 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | fnopabg 6706 | . 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 1537 ∈ wcel 2106 ∃!weu 2566 ∀wral 3059 {copab 5210 Fn wfn 6558 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-fun 6565 df-fn 6566 |
This theorem is referenced by: fvopab3g 7011 |
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