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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 1971 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦) | |
2 | nfeu1 1952 | . . . . . 6 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 | |
3 | nfv 1461 | . . . . . 6 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴) | |
4 | 2, 3 | nfim 1504 | . . . . 5 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
5 | tz6.12-1 5221 | . . . . . . . 8 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
6 | 5 | expcom 114 | . . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦)) |
7 | breq2 3789 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦)) | |
8 | 7 | biimprd 156 | . . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
9 | 6, 8 | syli 37 | . . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
10 | 9 | com12 30 | . . . . 5 ⊢ (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
11 | 4, 10 | exlimi 1525 | . . . 4 ⊢ (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
12 | 1, 11 | mpcom 36 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
13 | 12, 7 | syl5ibcom 153 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦)) |
14 | 13, 6 | impbid 127 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∃wex 1421 ∃!weu 1941 class class class wbr 3785 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 |
This theorem is referenced by: fnbrfvb 5235 |
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