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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 5257 | . 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 5247 | . 2 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ {𝑦 ∣ 𝜑} = {𝑦}} |
4 | euabsn 4205 | . . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}) | |
5 | 4 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}} |
6 | 1, 3, 5 | 3eqtr4i 2642 | 1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 ∃!weu 2458 {cab 2596 {csn 4125 {copab 4642 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-dm 5048 |
This theorem is referenced by: opabiota 6171 |
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