Theorem dfafv2 47846

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 784	. . . . . 6 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3		reuaiotaiota 47802	. . . . . 6 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
4	2, 3	sylib 221	. . . . 5 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
5	1, 4	eqtrid 2810	. . . 4 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹‘𝐴) = (℩'𝑥𝐴𝐹𝑥))
6		eubrdm 47750	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹)
7	6	ancri 558	. . . . . . . 8 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
8	7	con3i 155	. . . . . . 7 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥)
9	8	adantl 486	. . . . . 6 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥)
10		aiotavb 47804	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
11	9, 10	sylib 221	. . . . 5 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
12	11	eqcomd 2769	. . . 4 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
13	5, 12	ifeqda 4525	. . 3 ⊢ (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
14	13	mptru 1577	. 2 ⊢ if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15		dfdfat2 47842	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16		ifbi 4511	. . 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 47834	. 2 ⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
19	14, 17, 18	3eqtr4ri 2797	1 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  ∃!weu 2596  Vcvv 3455  ifcif 4488   class class class wbr 5110  dom cdm 5663  ℩cio 6492  ‘cfv 6538  ℩'caiota 47797   defAt wdfat 47830  '''cafv 47831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6494  df-fun 6540  df-fv 6546  df-aiota 47799  df-dfat 47833  df-afv 47834

This theorem is referenced by:  afveq12d  47847  nfafv  47850  afvfundmfveq  47852  afvnfundmuv  47853  afvpcfv0  47860