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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 5777 | . 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 5766 | . 2 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
4 | euabsn 4654 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
5 | 4 | abbii 2883 | . 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}} |
6 | 1, 3, 5 | 3eqtr4i 2851 | 1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∃wex 1771 ∃!weu 2646 {cab 2796 {csn 4557 {copab 5119 dom cdm 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-dm 5558 |
This theorem is referenced by: opabiota 6739 |
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