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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 2620 | . . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
2 | 1 | albii 1827 | . . . . 5 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) |
3 | funopab 6393 | . . . . 5 ⊢ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-ral 3056 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑)) | |
5 | 2, 3, 4 | 3bitr4ri 307 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
6 | dmopab3 5773 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴) | |
7 | 5, 6 | anbi12i 630 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑) ↔ (Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)) |
8 | r19.26 3082 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑)) | |
9 | df-fn 6361 | . . 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 306 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴) |
11 | df-eu 2568 | . . . 4 ⊢ (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑)) | |
12 | 11 | biancomi 466 | . . 3 ⊢ (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑)) |
13 | 12 | ralbii 3078 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑)) |
14 | fnopabg.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
15 | 14 | fneq1i 6454 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴) |
16 | 10, 13, 15 | 3bitr4i 306 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∃*wmo 2537 ∃!weu 2567 ∀wral 3051 {copab 5101 dom cdm 5536 Fun wfun 6352 Fn wfn 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-fun 6360 df-fn 6361 |
This theorem is referenced by: fnopab 6494 mptfng 6495 axcontlem2 27010 |
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