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Mirrors > Home > ILE Home > Th. List > dffv3g | GIF version |
Description: A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
dffv3g | ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 5246 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | vex 2755 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | elimasng 5017 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 〈𝐴, 𝑥〉 ∈ 𝐹)) | |
4 | df-br 4022 | . . . . 5 ⊢ (𝐴𝐹𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐹) | |
5 | 3, 4 | bitr4di 198 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
6 | 2, 5 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥)) |
7 | 6 | iotabidv 5221 | . 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥)) |
8 | 1, 7 | eqtr4id 2241 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {csn 3610 〈cop 3613 class class class wbr 4021 “ cima 4650 ℩cio 5197 ‘cfv 5238 |
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-uni 3828 df-br 4022 df-opab 4083 df-xp 4653 df-cnv 4655 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fv 5246 |
This theorem is referenced by: dffv4g 5534 fvco2 5609 shftval 10875 |
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