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Mirrors > Home > ILE Home > Th. List > tz6.12c | GIF version |
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
tz6.12c | ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 1930 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦) | |
2 | nfeu1 1911 | . . . . . 6 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 | |
3 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴) | |
4 | 2, 3 | nfim 1464 | . . . . 5 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
5 | tz6.12-1 5200 | . . . . . . . 8 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
6 | 5 | expcom 109 | . . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦)) |
7 | breq2 3768 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦)) | |
8 | 7 | biimprd 147 | . . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
9 | 6, 8 | syli 33 | . . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
10 | 9 | com12 27 | . . . . 5 ⊢ (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
11 | 4, 10 | exlimi 1485 | . . . 4 ⊢ (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
12 | 1, 11 | mpcom 32 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
13 | 12, 7 | syl5ibcom 144 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦)) |
14 | 13, 6 | impbid 120 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∃wex 1381 ∃!weu 1900 class class class wbr 3764 ‘cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 |
This theorem is referenced by: fnbrfvb 5214 |
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