Theorem afveu 47341

Description: The value of a function at a unique point, analogous to fveu 6821. (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 5097	. . . 4 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
2	1	eubii 2583	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹)
3		eu2ndop1stv 47313	. . 3 ⊢ (∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹 → 𝐴 ∈ V)
4	2, 3	sylbi 217	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ V)
5		euex 2575	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6		eldmg 5845	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
7	5, 6	syl5ibrcom 247	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
8	7	impcom 407	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
9		dfdfat2 47316	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10		afvfundmfveq 47326	. . . . . . . . 9 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
11		fveu 6821	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
12	10, 11	sylan9eq 2789	. . . . . . . 8 ⊢ ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
13	12	ex 412	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
14	9, 13	sylbir 235	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
15	14	expcom 413	. . . . 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 481	. . 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 688	1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2566  {cab 2712  Vcvv 3438  ⟨cop 4584  ∪ cuni 4861   class class class wbr 5096  dom cdm 5622  ‘cfv 6490   defAt wdfat 47304  '''cafv 47305

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-aiota 47273  df-dfat 47307  df-afv 47308

This theorem is referenced by: (None)