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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tz6.12c-afv2 | Structured version Visualization version GIF version | ||
| Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6928. (Contributed by AV, 5-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| tz6.12c-afv2 | ⊢ (∃!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-afv2 47253 | . . . . . 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 ''''cafv2 47220 | 
| 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 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| 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-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-dfat 47131 df-afv2 47221 | 
| This theorem is referenced by: tz6.12i-afv2 47255 dfatbrafv2b 47257 fnbrafv2b 47260 dfatcolem 47267 | 
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