Theorem dfafv2 44624

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 6441	. . . . 5 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		simprr 770	. . . . . 6 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3		reuaiotaiota 44580	. . . . . 6 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
4	2, 3	sylib 217	. . . . 5 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
5	1, 4	eqtrid 2790	. . . 4 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹‘𝐴) = (℩'𝑥𝐴𝐹𝑥))
6		eubrdm 44530	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹)
7	6	ancri 550	. . . . . . . 8 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
8	7	con3i 154	. . . . . . 7 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥)
9	8	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥)
10		aiotavb 44582	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
11	9, 10	sylib 217	. . . . 5 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
12	11	eqcomd 2744	. . . 4 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
13	5, 12	ifeqda 4495	. . 3 ⊢ (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
14	13	mptru 1546	. 2 ⊢ if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15		dfdfat2 44620	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16		ifbi 4481	. . 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 44612	. 2 ⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
19	14, 17, 18	3eqtr4ri 2777	1 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432  ifcif 4459   class class class wbr 5074  dom cdm 5589  ℩cio 6389  ‘cfv 6433  ℩'caiota 44575   defAt wdfat 44608  '''cafv 44609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612

This theorem is referenced by:  afveq12d  44625  nfafv  44628  afvfundmfveq  44630  afvnfundmuv  44631  afvpcfv0  44638