![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fnopabg | Structured version Visualization version GIF version |
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) |
Ref | Expression |
---|---|
fnopabg.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Ref | Expression |
---|---|
fnopabg | ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moanimv 2617 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
2 | 1 | albii 1816 | . . . . 5 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) |
3 | funopab 6603 | . . . . 5 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-ral 3060 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
5 | 2, 3, 4 | 3bitr4ri 304 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
6 | dmopab3 5933 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴) | |
7 | 5, 6 | anbi12i 628 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑) ↔ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)) |
8 | r19.26 3109 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑)) | |
9 | df-fn 6566 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ↔ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)) | |
10 | 7, 8, 9 | 3bitr4i 303 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴) |
11 | df-eu 2567 | . . . 4 ⊢ (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑)) | |
12 | 11 | biancomi 462 | . . 3 ⊢ (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑)) |
13 | 12 | ralbii 3091 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑)) |
14 | fnopabg.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
15 | 14 | fneq1i 6666 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴) |
16 | 10, 13, 15 | 3bitr4i 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 ∃!weu 2566 ∀wral 3059 {copab 5210 dom cdm 5689 Fun wfun 6557 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: fnopab 6707 mptfng 6708 axcontlem2 28995 tfsconcatfn 43328 tfsconcatfv1 43329 tfsconcatfv2 43330 |
Copyright terms: Public domain | W3C validator |