Theorem dfafv2 48295

Description: Alternative definition of (𝐹'''𝐴) using (𝐹‘𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.)

Assertion

Ref	Expression
dfafv2	⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)

Proof of Theorem dfafv2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 6546	. . . . 5 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		simprr 785	. . . . . 6 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3		reuaiotaiota 48251	. . . . . 6 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
4	2, 3	sylib 221	. . . . 5 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
5	1, 4	eqtrid 2807	. . . 4 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹‘𝐴) = (℩'𝑥𝐴𝐹𝑥))
6		eubrdm 48199	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹)
7	6	ancri 559	. . . . . . . 8 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
8	7	con3i 155	. . . . . . 7 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥)
9	8	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥)
10		aiotavb 48253	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
11	9, 10	sylib 221	. . . . 5 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
12	11	eqcomd 2766	. . . 4 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
13	5, 12	ifeqda 4518	. . 3 ⊢ (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
14	13	mptru 1576	. 2 ⊢ if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15		dfdfat2 48291	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16		ifbi 4504	. . 3 ⊢ ((𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V))
17	15, 16	ax-mp 5	. 2 ⊢ if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V)
18		df-afv 48283	. 2 ⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
19	14, 17, 18	3eqtr4ri 2794	1 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  ∃!weu 2593  Vcvv 3450  ifcif 4481   class class class wbr 5102  dom cdm 5650  ℩cio 6492  ‘cfv 6538  ℩'caiota 48246   defAt wdfat 48279  '''cafv 48280

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-11 2193  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-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  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-int 4907  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-fv 6546  df-aiota 48248  df-dfat 48282  df-afv 48283

This theorem is used by:  afveq12d  48296  nfafv  48299  afvfundmfveq  48301  afvnfundmuv  48302  afvpcfv0  48309