Theorem afveu 47158

Description: The value of a function at a unique point, analogous to fveu 6850. (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 5111	. . . 4 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
2	1	eubii 2579	. . 3 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹)
3		eu2ndop1stv 47130	. . 3 ⊢ (∃!𝑥⟨𝐴, 𝑥⟩ ∈ 𝐹 → 𝐴 ∈ V)
4	2, 3	sylbi 217	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ V)
5		euex 2571	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6		eldmg 5865	. . . . 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 47133	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10		afvfundmfveq 47143	. . . . . . . . 9 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
11		fveu 6850	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
12	10, 11	sylan9eq 2785	. . . . . . . 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 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2562  {cab 2708  Vcvv 3450  ⟨cop 4598  ∪ cuni 4874   class class class wbr 5110  dom cdm 5641  ‘cfv 6514   defAt wdfat 47121  '''cafv 47122

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-aiota 47090  df-dfat 47124  df-afv 47125

This theorem is referenced by: (None)