Theorem dfatbrafv2b 44624

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6804 or funbrfvb 6806. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 44592). (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 2738	. . . 4 ⊢ (𝐹''''𝐴) = (𝐹''''𝐴)
2		dfatafv2ex 44592	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
3	2	adantr 480	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → (𝐹''''𝐴) ∈ V)
4		eqeq2 2750	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐴) → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = (𝐹''''𝐴)))
5		breq2 5074	. . . . . . 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 44507	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9		tz6.12c-afv2 44621	. . . . . . 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 3484	. . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = (𝐹''''𝐴) ↔ 𝐴𝐹(𝐹''''𝐴)))
13	1, 12	mpbii 232	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → 𝐴𝐹(𝐹''''𝐴))
14		breq2 5074	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
15	13, 14	syl5ibcom 244	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))
16		df-dfat 44498	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
17		simpll 763	. . . . 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 44623	. . 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 211	1 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃!weu 2568  Vcvv 3422  {csn 4558   class class class wbr 5070  dom cdm 5580   ↾ cres 5582  Fun wfun 6412   defAt wdfat 44495  ''''cafv2 44587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fn 6421  df-dfat 44498  df-afv2 44588

This theorem is referenced by:  dfatopafv2b  44625  dfatsnafv2  44631  dfatcolem  44634