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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfafv2 | Structured version Visualization version GIF version |
Description: Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.) |
Ref | Expression |
---|---|
dfafv2 | ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 6541 | . . . . 5 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | simprr 770 | . . . . . 6 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥) | |
3 | reuaiotaiota 46281 | . . . . . 6 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥)) | |
4 | 2, 3 | sylib 217 | . . . . 5 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥)) |
5 | 1, 4 | eqtrid 2776 | . . . 4 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹‘𝐴) = (℩'𝑥𝐴𝐹𝑥)) |
6 | eubrdm 46231 | . . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹) | |
7 | 6 | ancri 549 | . . . . . . . 8 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) |
8 | 7 | con3i 154 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥) |
9 | 8 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥) |
10 | aiotavb 46283 | . . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V) | |
11 | 9, 10 | sylib 217 | . . . . 5 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V) |
12 | 11 | eqcomd 2730 | . . . 4 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥)) |
13 | 5, 12 | ifeqda 4556 | . . 3 ⊢ (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥)) |
14 | 13 | mptru 1540 | . 2 ⊢ if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥) |
15 | dfdfat2 46321 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) | |
16 | ifbi 4542 | . . 3 ⊢ ((𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V)) | |
17 | 15, 16 | ax-mp 5 | . 2 ⊢ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) |
18 | df-afv 46313 | . 2 ⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥) | |
19 | 14, 17, 18 | 3eqtr4ri 2763 | 1 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∃!weu 2554 Vcvv 3466 ifcif 4520 class class class wbr 5138 dom cdm 5666 ℩cio 6483 ‘cfv 6533 ℩'caiota 46276 defAt wdfat 46309 '''cafv 46310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-res 5678 df-iota 6485 df-fun 6535 df-fv 6541 df-aiota 46278 df-dfat 46312 df-afv 46313 |
This theorem is referenced by: afveq12d 46326 nfafv 46329 afvfundmfveq 46331 afvnfundmuv 46332 afvpcfv0 46339 |
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