Theorem bj-fvimacnv0 36471

Description: Variant of fvimacnv 7054 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 46127. (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 2820	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ∅ ∈ 𝐵))
2	1	biimpcd 248	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → ((𝐹‘𝐴) = ∅ → ∅ ∈ 𝐵))
3	2	con3rr3 155	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → ¬ (𝐹‘𝐴) = ∅))
4	3	imp 406	. . . . . . 7 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → ¬ (𝐹‘𝐴) = ∅)
5		ndmfv 6926	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
6	4, 5	nsyl2 141	. . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → 𝐴 ∈ dom 𝐹)
7		simpr 484	. . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)
8		fvimacnv 7054	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
9	8	biimpd 228	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
10	9	ex 412	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
11	10	com3l 89	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝐵 → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
12	6, 7, 11	sylc 65	. . . . 5 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
13	12	ex 412	. . . 4 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → (Fun 𝐹 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
14	13	com3r 87	. . 3 ⊢ (Fun 𝐹 → (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵))))
15	14	imp 406	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
16		fvimacnvi 7053	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵)
17	16	ex 412	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
18	17	adantr 480	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
19	15, 18	impbid 211	1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∅c0 4322  ^◡ccnv 5675  dom cdm 5676   “ cima 5679  Fun wfun 6537  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by: (None)