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| Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.) | 
| Ref | Expression | 
|---|---|
| opabiota.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | 
| Ref | Expression | 
|---|---|
| opabiotadm | ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dmopab 5925 | . 2 ⊢ dom {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}} | |
| 2 | opabiota.1 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | |
| 3 | 2 | dmeqi 5914 | . 2 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} | 
| 4 | euabsn 4725 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
| 5 | 4 | abbii 2808 | . 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}} | 
| 6 | 1, 3, 5 | 3eqtr4i 2774 | 1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∃wex 1778 ∃!weu 2567 {cab 2713 {csn 4625 {copab 5204 dom cdm 5684 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-dm 5694 | 
| This theorem is referenced by: opabiota 6990 | 
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