Theorem bj-fvimacnv0 37987

Description: Variant of fvimacnv 7052 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with Definition df-afv 47915. (Contributed by BJ, 7-Jan-2024.)

Assertion

Ref	Expression
bj-fvimacnv0	⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Proof of Theorem bj-fvimacnv0

Step	Hyp	Ref	Expression
1		eleq1 2853	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ∅ ∈ 𝐵))
2	1	biimpcd 252	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → ((𝐹‘𝐴) = ∅ → ∅ ∈ 𝐵))
3	2	con3rr3 156	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → ¬ (𝐹‘𝐴) = ∅))
4	3	imp 412	. . . . . . 7 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → ¬ (𝐹‘𝐴) = ∅)
5		ndmfv 6917	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
6	4, 5	nsyl2 142	. . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → 𝐴 ∈ dom 𝐹)
7		simpr 490	. . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)
8		fvimacnv 7052	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
9	8	biimpd 232	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
10	9	ex 418	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
11	10	com3l 90	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝐵 → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
12	6, 7, 11	sylc 66	. . . . 5 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
13	12	ex 418	. . . 4 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
14	13	com3r 88	. . 3 ⊢ (Fun 𝐹 → (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
15	14	imp 412	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
16		fvimacnvi 7051	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵)
17	16	ex 418	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
18	17	adantr 486	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
19	15, 18	impbid 215	1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∅c0 4286  ^◡ccnv 5662  dom cdm 5663   “ cima 5666  Fun wfun 6534  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is used by: (None)