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

Theorem intrnfi 8733
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 1187 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴𝐵)
21frnd 6396 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹𝐵)
31fdmd 6398 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 = 𝐴)
4 simpr2 1188 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
53, 4eqnetrd 3053 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 ≠ ∅)
6 dm0rn0 5686 . . . . 5 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
76necon3bii 3038 . . . 4 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
85, 7sylib 219 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ≠ ∅)
9 simpr3 1189 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
101ffnd 6390 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹 Fn 𝐴)
11 dffn4 6471 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1210, 11sylib 219 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴onto→ran 𝐹)
13 fofi 8663 . . . 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 1121 . 2 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin))
16 elfir 8732 . 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 396  w3a 1080  wcel 2083  wne 2986  wss 3865  c0 4217   cint 4788  dom cdm 5450  ran crn 5451   Fn wfn 6227  wf 6228  ontowfo 6230  cfv 6232  Fincfn 8364  ficfi 8727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-om 7444  df-1o 7960  df-er 8146  df-en 8365  df-dom 8366  df-fin 8368  df-fi 8728
This theorem is referenced by:  iinfi  8734  firest  16539
  Copyright terms: Public domain W3C validator