Theorem fimacnvdisj 6785

Description: The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
fimacnvdisj	⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅)

Proof of Theorem fimacnvdisj

Step	Hyp	Ref	Expression
1		df-rn 5695	. . . 4 ⊢ ran 𝐹 = dom ◡𝐹
2		frn 6742	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3	2	adantr 480	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → ran 𝐹 ⊆ 𝐵)
4	1, 3	eqsstrrid 4022	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → dom ◡𝐹 ⊆ 𝐵)
5		ssdisj 4459	. . 3 ⊢ ((dom ◡𝐹 ⊆ 𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅)
6	4, 5	sylancom 588	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (dom ◡𝐹 ∩ 𝐶) = ∅)
7		imadisj 6097	. 2 ⊢ ((◡𝐹 “ 𝐶) = ∅ ↔ (dom ◡𝐹 ∩ 𝐶) = ∅)
8	6, 7	sylibr 234	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   “ cima 5687  ⟶wf 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-f 6564

This theorem is referenced by:  vdwmc2  17018  gsumval3a  19922  psrbag0  22087  mbfconstlem  25663  itg1val2  25720  ofpreima2  32677