Theorem afveu 47616

Description: The value of a function at a unique point, analogous to fveu 6816. (Contributed by Alexander van der Vekens, 29-Nov-2017.)

Assertion

Ref	Expression
afveu	⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem afveu

Step	Hyp	Ref	Expression
1		df-br 5073	. . . 4 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
2	1	eubii 2589	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹)
3		eu2ndop1stv 47588	. . 3 ⊢ (∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹 → 𝐴 ∈ V)
4	2, 3	sylbi 218	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ V)
5		euex 2581	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6		eldmg 5840	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
7	5, 6	syl5ibrcom 248	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
8	7	impcom 408	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
9		dfdfat2 47591	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10		afvfundmfveq 47601	. . . . . . . . 9 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
11		fveu 6816	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
12	10, 11	sylan9eq 2794	. . . . . . . 8 ⊢ ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
13	12	ex 413	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
14	9, 13	sylbir 236	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
15	14	expcom 414	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})))
16	15	pm2.43a 54	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
17	16	adantl 482	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
18	8, 17	mpd 15	. 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
19	4, 18	mpancom 694	1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572  {cab 2717  Vcvv 3431  ⟨cop 4561  ∪ cuni 4838   class class class wbr 5072  dom cdm 5618  ‘cfv 6485   defAt wdfat 47579  '''cafv 47580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-aiota 47548  df-dfat 47582  df-afv 47583

This theorem is referenced by: (None)