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Theorem intrnfi 8877
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 1191 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴𝐵)
21frnd 6510 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹𝐵)
31fdmd 6513 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 = 𝐴)
4 simpr2 1192 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
53, 4eqnetrd 3081 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → dom 𝐹 ≠ ∅)
6 dm0rn0 5782 . . . . 5 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
76necon3bii 3066 . . . 4 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
85, 7sylib 221 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ≠ ∅)
9 simpr3 1193 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
101ffnd 6504 . . . . 5 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹 Fn 𝐴)
11 dffn4 6587 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1210, 11sylib 221 . . . 4 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐹:𝐴onto→ran 𝐹)
13 fofi 8807 . . . 4 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto→ran 𝐹) → ran 𝐹 ∈ Fin)
149, 12, 13syl2anc 587 . . 3 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ Fin)
152, 8, 143jca 1125 . 2 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin))
16 elfir 8876 . 2 ((𝐵𝑉 ∧ (ran 𝐹𝐵 ∧ ran 𝐹 ≠ ∅ ∧ ran 𝐹 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
1715, 16syldan 594 1 ((𝐵𝑉 ∧ (𝐹:𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ran 𝐹 ∈ (fi‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115  wne 3014  wss 3919  c0 4276   cint 4862  dom cdm 5542  ran crn 5543   Fn wfn 6338  wf 6339  ontowfo 6341  cfv 6343  Fincfn 8505  ficfi 8871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-1o 8098  df-er 8285  df-en 8506  df-dom 8507  df-fin 8509  df-fi 8872
This theorem is referenced by:  iinfi  8878  firest  16706
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