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Theorem funss 5207
Description: Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Assertion
Ref Expression
funss (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))

Proof of Theorem funss
StepHypRef Expression
1 relss 4691 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
2 coss1 4759 . . . . 5 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐴))
3 cnvss 4777 . . . . . 6 (𝐴𝐵𝐴𝐵)
4 coss2 4760 . . . . . 6 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
53, 4syl 14 . . . . 5 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
62, 5sstrd 3152 . . . 4 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐵))
7 sstr2 3149 . . . 4 ((𝐴𝐴) ⊆ (𝐵𝐵) → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
86, 7syl 14 . . 3 (𝐴𝐵 → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
91, 8anim12d 333 . 2 (𝐴𝐵 → ((Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I )))
10 df-fun 5190 . 2 (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ))
11 df-fun 5190 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
129, 10, 113imtr4g 204 1 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wss 3116   I cid 4266  ccnv 4603  ccom 4608  Rel wrel 4609  Fun wfun 5182
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-fun 5190
This theorem is referenced by:  funeq  5208  funopab4  5225  funres  5229  fun0  5246  funcnvcnv  5247  funin  5259  funres11  5260  foimacnv  5450  tfrlemibfn  6296  tfr1onlembfn  6312  tfrcllembfn  6325  strslssd  12440  strle1g  12485
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