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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 5929 | . 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 5918 | . 2 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
4 | euabsn 4731 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
5 | 4 | abbii 2807 | . 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}} |
6 | 1, 3, 5 | 3eqtr4i 2773 | 1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1776 ∃!weu 2566 {cab 2712 {csn 4631 {copab 5210 dom cdm 5689 |
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-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-dm 5699 |
This theorem is referenced by: opabiota 6991 |
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