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Mirrors > Home > ILE Home > Th. List > args | GIF version |
Description: Two ways to express the class of unique-valued arguments of 𝐹, which is the same as the domain of 𝐹 whenever 𝐹 is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg 𝐹 " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.) |
Ref | Expression |
---|---|
args | ⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2755 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | imasng 5014 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦}) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦} |
4 | 3 | eqeq1i 2197 | . . . 4 ⊢ ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦 ∣ 𝑥𝐹𝑦} = {𝑦}) |
5 | 4 | exbii 1616 | . . 3 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦}) |
6 | euabsn 3680 | . . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦}) | |
7 | 5, 6 | bitr4i 187 | . 2 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦) |
8 | 7 | abbii 2305 | 1 ⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∃wex 1503 ∃!weu 2038 ∈ wcel 2160 {cab 2175 Vcvv 2752 {csn 3610 class class class wbr 4021 “ cima 4650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-br 4022 df-opab 4083 df-xp 4653 df-cnv 4655 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 |
This theorem is referenced by: (None) |
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