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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 6695. (Contributed by AV, 5-Sep-2022.) |
Ref | Expression |
---|---|
tz6.12c-afv2 | ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeu1 2674 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 | |
2 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹''''𝐴) | |
3 | euex 2662 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦) | |
4 | tz6.12-1-afv2 43489 | . . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦) | |
5 | 4 | expcom 416 | . . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹''''𝐴) = 𝑦)) |
6 | breq2 5070 | . . . . . 6 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝑦)) | |
7 | 6 | biimprd 250 | . . . . 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 247 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 → 𝐴𝐹𝑦)) |
11 | 10, 5 | impbid 214 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∃!weu 2653 class class class wbr 5066 ''''cafv2 43456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-dfat 43367 df-afv2 43457 |
This theorem is referenced by: tz6.12i-afv2 43491 dfatbrafv2b 43493 fnbrafv2b 43496 dfatcolem 43503 |
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