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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  –onto→wfo 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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