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Theorem fintm 5352
 Description: Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
Hypothesis
Ref Expression
fintm.1 𝑥 𝑥𝐵
Assertion
Ref Expression
fintm (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fintm
StepHypRef Expression
1 ssint 3823 . . . 4 (ran 𝐹 𝐵 ↔ ∀𝑥𝐵 ran 𝐹𝑥)
21anbi2i 453 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥))
3 fintm.1 . . . 4 𝑥 𝑥𝐵
4 r19.28mv 3486 . . . 4 (∃𝑥 𝑥𝐵 → (∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥)))
53, 4ax-mp 5 . . 3 (∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥))
62, 5bitr4i 186 . 2 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ ∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
7 df-f 5171 . 2 (𝐹:𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵))
8 df-f 5171 . . 3 (𝐹:𝐴𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
98ralbii 2463 . 2 (∀𝑥𝐵 𝐹:𝐴𝑥 ↔ ∀𝑥𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
106, 7, 93bitr4i 211 1 (𝐹:𝐴 𝐵 ↔ ∀𝑥𝐵 𝐹:𝐴𝑥)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1472   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  ∩ cint 3807  ran crn 4584   Fn wfn 5162  ⟶wf 5163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3808  df-f 5171 This theorem is referenced by: (None)
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