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

Theorem intrnfi 9295
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 1195 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴𝐵)
21frnd 6655 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹𝐵)
31fdmd 6657 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 = 𝐴)
4 simpr2 1196 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
53, 4eqnetrd 2993 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 ≠ ∅)
6 dm0rn0 5862 . . . . 5 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
76necon3bii 2978 . . . 4 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
85, 7sylib 218 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ≠ ∅)
9 simpr3 1197 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
101ffnd 6648 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹 Fn 𝐴)
11 dffn4 6737 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1210, 11sylib 218 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴onto→ran 𝐹)
13 fofi 9192 . . . 4 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto→ran 𝐹) → ran 𝐹 ∈ Fin)
149, 12, 13syl2anc 584 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ Fin)
152, 8, 143jca 1128 . 2 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin))
16 elfir 9294 . 2 ((𝐵𝑉 ∧ (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
1715, 16syldan 591 1 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2110  wne 2926  wss 3900  c0 4281   cint 4895  dom cdm 5614  ran crn 5615   Fn wfn 6472  wf 6473  ontowfo 6475  cfv 6477  Fincfn 8864  ficfi 9289
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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-om 7792  df-1o 8380  df-en 8865  df-dom 8866  df-fin 8868  df-fi 9290
This theorem is referenced by:  iinfi  9296  firest  17328
  Copyright terms: Public domain W3C validator