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Theorem fintm 5278
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 3757 . . . 4 (ran 𝐹 𝐵 ↔ ∀𝑥𝐵 ran 𝐹𝑥)
21anbi2i 452 . . 3 ((𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐵 ran 𝐹𝑥))
3 fintm.1 . . . 4 𝑥 𝑥𝐵
4 r19.28mv 3425 . . . 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 5097 . 2 (𝐹:𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 𝐵))
8 df-f 5097 . . 3 (𝐹:𝐴𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝑥))
98ralbii 2418 . 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 1453  wcel 1465  wral 2393  wss 3041   cint 3741  ran crn 4510   Fn wfn 5088  wf 5089
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-int 3742  df-f 5097
This theorem is referenced by: (None)
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