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Mirrors > Home > MPE Home > Th. List > opabiotadm | Structured version Visualization version GIF version |
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 5909 | . 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 5898 | . 2 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
4 | euabsn 4725 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
5 | 4 | abbii 2796 | . 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}} |
6 | 1, 3, 5 | 3eqtr4i 2764 | 1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1773 ∃!weu 2556 {cab 2703 {csn 4623 {copab 5203 dom cdm 5669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-dm 5679 |
This theorem is referenced by: opabiota 6968 |
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