Theorem afv2eu 48401

Description: The value of a function at a unique point, analogous to fveu 6874. (Contributed by AV, 5-Sep-2022.)

Assertion

Ref	Expression
afv2eu	⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem afv2eu

Step	Hyp	Ref	Expression
1		eubrv 48198	. 2 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ V)
2		euex 2602	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
3		eldmg 5879	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
4	2, 3	syl5ibrcom 250	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
5	4	impcom 413	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
6		dfdfat2 48291	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
7		dfatafv2iota 48373	. . . . . . . . 9 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
8		iotauni 6515	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
9	7, 8	sylan9eq 2815	. . . . . . . 8 ⊢ ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
10	9	ex 418	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
11	6, 10	sylbir 238	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
12	11	expcom 419	. . . . 5 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})))
13	12	pm2.43a 55	. . . 4 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
14	13	adantl 487	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}))
15	5, 14	mpd 16	. 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
16	1, 15	mpancom 701	1 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃!weu 2593  {cab 2738  Vcvv 3450  ∪ cuni 4866   class class class wbr 5102  dom cdm 5650  ℩cio 6492   defAt wdfat 48279  ''''cafv2 48371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-dfat 48282  df-afv2 48372

This theorem is used by: (None)