Theorem dfatbrafv2b 47160

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6973 or funbrfvb 6975. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 47128). (Contributed by AV, 6-Sep-2022.)

Assertion

Ref	Expression
dfatbrafv2b	⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))

Proof of Theorem dfatbrafv2b

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . 4 ⊢ (𝐹''''𝐴) = (𝐹''''𝐴)
2		dfatafv2ex 47128	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
3	2	adantr 480	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → (𝐹''''𝐴) ∈ V)
4		eqeq2 2752	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐴) → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = (𝐹''''𝐴)))
5		breq2 5170	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐴) → (𝐴𝐹𝑥 ↔ 𝐴𝐹(𝐹''''𝐴)))
6	4, 5	bibi12d 345	. . . . . 6 ⊢ (𝑥 = (𝐹''''𝐴) → (((𝐹''''𝐴) = 𝑥 ↔ 𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
7	6	adantl 481	. . . . 5 ⊢ (((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) ∧ 𝑥 = (𝐹''''𝐴)) → (((𝐹''''𝐴) = 𝑥 ↔ 𝐴𝐹𝑥) ↔ ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴))))
8		dfdfat2 47043	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9		tz6.12c-afv2 47157	. . . . . . 7 ⊢ (∃!𝑥 𝐴𝐹𝑥 → ((𝐹''''𝐴) = 𝑥 ↔ 𝐴𝐹𝑥))
10	8, 9	simplbiim 504	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑥 ↔ 𝐴𝐹𝑥))
11	10	adantr 480	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝑥 ↔ 𝐴𝐹𝑥))
12	3, 7, 11	vtocld 3573	. . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴)))
13	1, 12	mpbii 233	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → 𝐴𝐹(𝐹''''𝐴))
14		breq2 5170	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
15	13, 14	syl5ibcom 245	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))
16		df-dfat 47034	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
17		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ dom 𝐹)
18		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
19		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → Fun (𝐹 ↾ {𝐴}))
20	19	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵 ∈ 𝑊) → Fun (𝐹 ↾ {𝐴}))
21	17, 18, 20	jca31 514	. . . 4 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
22	16, 21	sylanb 580	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})))
23		funressnbrafv2 47159	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
24	22, 23	syl 17	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
25	15, 24	impbid 212	1 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  Vcvv 3488  {csn 4648   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  Fun wfun 6567   defAt wdfat 47031  ''''cafv2 47123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-dfat 47034  df-afv2 47124

This theorem is referenced by:  dfatopafv2b  47161  dfatsnafv2  47167  dfatcolem  47170