Theorem bj-fvimacnv0 34571

Description: Variant of fvimacnv 6823 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 43368. (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 2900	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹‘𝐴) ∈ 𝐵 ↔ ∅ ∈ 𝐵))
2	1	biimpcd 251	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → ((𝐹‘𝐴) = ∅ → ∅ ∈ 𝐵))
3	2	con3rr3 158	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐵 → ((𝐹‘𝐴) ∈ 𝐵 → ¬ (𝐹‘𝐴) = ∅))
4	3	imp 409	. . . . . . 7 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → ¬ (𝐹‘𝐴) = ∅)
5		ndmfv 6700	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
6	4, 5	nsyl2 143	. . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → 𝐴 ∈ dom 𝐹)
7		simpr 487	. . . . . 6 ⊢ ((¬ ∅ ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝐵) → (𝐹‘𝐴) ∈ 𝐵)
8		fvimacnv 6823	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
9	8	biimpd 231	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
10	9	ex 415	. . . . . . 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 415	. . . 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 409	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
16		fvimacnvi 6822	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵)
17	16	ex 415	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
18	17	adantr 483	. 2 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
19	15, 18	impbid 214	1 ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∅c0 4291  ^◡ccnv 5554  dom cdm 5555   “ cima 5558  Fun wfun 6349  ‘cfv 6355

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363

This theorem is referenced by:  bj-isrvec  34578