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Mirrors > Home > MPE Home > Th. List > tz6.12c | Structured version Visualization version 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 2482 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦) | |
2 | nfeu1 2468 | . . . . . 6 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 | |
3 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴) | |
4 | 2, 3 | nfim 1813 | . . . . 5 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
5 | tz6.12-1 6120 | . . . . . . . 8 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
6 | 5 | expcom 450 | . . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦)) |
7 | breq2 4587 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦)) | |
8 | 7 | biimprd 237 | . . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
9 | 6, 8 | syli 38 | . . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
10 | 9 | com12 32 | . . . . 5 ⊢ (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
11 | 4, 10 | exlimi 2073 | . . . 4 ⊢ (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
12 | 1, 11 | mpcom 37 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
13 | 12, 7 | syl5ibcom 234 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦)) |
14 | 13, 6 | impbid 201 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∃wex 1695 ∃!weu 2458 class class class wbr 4583 ‘cfv 5804 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: tz6.12i 6124 fnbrfvb 6146 |
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