Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2eu | Structured version Visualization version GIF version |
Description: The value of a function at a unique point, analogous to fveu 6658. (Contributed by AV, 5-Sep-2022.) |
Ref | Expression |
---|---|
afv2eu | ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eubrv 43344 | . 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ V) | |
2 | euex 2661 | . . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥) | |
3 | eldmg 5764 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥)) | |
4 | 2, 3 | syl5ibrcom 249 | . . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹)) |
5 | 4 | impcom 410 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹) |
6 | dfdfat2 43401 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) | |
7 | dfatafv2iota 43483 | . . . . . . . . 9 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | |
8 | iotauni 6327 | . . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | |
9 | 7, 8 | sylan9eq 2875 | . . . . . . . 8 ⊢ ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) |
10 | 9 | ex 415 | . . . . . . 7 ⊢ (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})) |
11 | 6, 10 | sylbir 237 | . . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})) |
12 | 11 | expcom 416 | . . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))) |
13 | 12 | pm2.43a 54 | . . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})) |
14 | 13 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})) |
15 | 5, 14 | mpd 15 | . 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) |
16 | 1, 15 | mpancom 686 | 1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ∃!weu 2652 {cab 2798 Vcvv 3493 ∪ cuni 4835 class class class wbr 5063 dom cdm 5552 ℩cio 6309 defAt wdfat 43389 ''''cafv2 43481 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-res 5564 df-iota 6311 df-fun 6354 df-dfat 43392 df-afv2 43482 |
This theorem is referenced by: (None) |
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