Theorem afveu 41233

Description: The value of a function at a unique point, analogous to fveu 6183. (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 4654	. . . 4 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
2	1	eubii 2492	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹)
3		eu2ndop1stv 41202	. . 3 ⊢ (∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹 → 𝐴 ∈ V)
4	2, 3	sylbi 207	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ V)
5		euex 2494	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6		eldmg 5319	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
7	5, 6	syl5ibrcom 237	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
8	7	impcom 446	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
9		dfdfat2 41211	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10		afvfundmfveq 41218	. . . . . . . . 9 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
11		fveu 6183	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
12	10, 11	sylan9eq 2676	. . . . . . . 8 ⊢ ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
13	12	ex 450	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
14	9, 13	sylbir 225	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
15	14	expcom 451	. . . . 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 703	1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  {cab 2608  Vcvv 3200  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114  ‘cfv 5888   defAt wdfat 41193  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by: (None)