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| Mirrors > Home > MPE Home > Th. List > tz6.12cOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of tz6.12c 6928 as of 23-Dec-2024. (Contributed by NM, 30-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| tz6.12cOLD | ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfeu1 2588 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 | |
| 2 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴) | |
| 3 | euex 2577 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦) | |
| 4 | tz6.12-1 6929 | . . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
| 5 | 4 | expcom 413 | . . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦)) |
| 6 | breq2 5147 | . . . . . 6 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦)) | |
| 7 | 6 | biimprd 248 | . . . . 5 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
| 8 | 5, 7 | syli 39 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
| 9 | 1, 2, 3, 8 | exlimimdd 2219 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
| 10 | 9, 6 | syl5ibcom 245 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦)) |
| 11 | 10, 5 | impbid 212 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃!weu 2568 class class class wbr 5143 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: (None) |
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