Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  intrnfi Structured version   Visualization version   GIF version

Theorem intrnfi 8205
 Description: Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
intrnfi ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))

Proof of Theorem intrnfi
StepHypRef Expression
1 simpr1 1060 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴𝐵)
2 frn 5966 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
31, 2syl 17 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹𝐵)
4 fdm 5964 . . . . . 6 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
51, 4syl 17 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 = 𝐴)
6 simpr2 1061 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
75, 6eqnetrd 2849 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 ≠ ∅)
8 dm0rn0 5263 . . . . 5 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
98necon3bii 2834 . . . 4 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
107, 9sylib 207 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ≠ ∅)
11 simpr3 1062 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
12 ffn 5958 . . . . . 6 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
131, 12syl 17 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹 Fn 𝐴)
14 dffn4 6034 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1513, 14sylib 207 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴onto→ran 𝐹)
16 fofi 8135 . . . 4 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto→ran 𝐹) → ran 𝐹 ∈ Fin)
1711, 15, 16syl2anc 691 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ Fin)
183, 10, 173jca 1235 . 2 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin))
19 elfir 8204 . 2 ((𝐵𝑉 ∧ (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
2018, 19syldan 486 1 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  Fincfn 7841  ficfi 8199 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200 This theorem is referenced by:  iinfi  8206  firest  15916
 Copyright terms: Public domain W3C validator