Theorem dfafv2 47592

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 6500	. . . . 5 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		simprr 773	. . . . . 6 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ∃!𝑥 𝐴𝐹𝑥)
3		reuaiotaiota 47548	. . . . . 6 ⊢ (∃!𝑥 𝐴𝐹𝑥 ↔ (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
4	2, 3	sylib 218	. . . . 5 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩𝑥𝐴𝐹𝑥) = (℩'𝑥𝐴𝐹𝑥))
5	1, 4	eqtrid 2784	. . . 4 ⊢ ((⊤ ∧ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (𝐹‘𝐴) = (℩'𝑥𝐴𝐹𝑥))
6		eubrdm 47496	. . . . . . . . 9 ⊢ (∃!𝑥 𝐴𝐹𝑥 → 𝐴 ∈ dom 𝐹)
7	6	ancri 549	. . . . . . . 8 ⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
8	7	con3i 154	. . . . . . 7 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → ¬ ∃!𝑥 𝐴𝐹𝑥)
9	8	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → ¬ ∃!𝑥 𝐴𝐹𝑥)
10		aiotavb 47550	. . . . . 6 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 ↔ (℩'𝑥𝐴𝐹𝑥) = V)
11	9, 10	sylib 218	. . . . 5 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → (℩'𝑥𝐴𝐹𝑥) = V)
12	11	eqcomd 2743	. . . 4 ⊢ ((⊤ ∧ ¬ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥)) → V = (℩'𝑥𝐴𝐹𝑥))
13	5, 12	ifeqda 4504	. . 3 ⊢ (⊤ → if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥))
14	13	mptru 1549	. 2 ⊢ if((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥), (𝐹‘𝐴), V) = (℩'𝑥𝐴𝐹𝑥)
15		dfdfat2 47588	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
16		ifbi 4490	. . 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 47580	. 2 ⊢ (𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
19	14, 17, 18	3eqtr4ri 2771	1 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∃!weu 2569  Vcvv 3430  ifcif 4467   class class class wbr 5086  dom cdm 5624  ℩cio 6446  ‘cfv 6492  ℩'caiota 47543   defAt wdfat 47576  '''cafv 47577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-aiota 47545  df-dfat 47579  df-afv 47580

This theorem is referenced by:  afveq12d  47593  nfafv  47596  afvfundmfveq  47598  afvnfundmuv  47599  afvpcfv0  47606