Theorem structfun 12493

Description: Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)

Hypothesis

Ref	Expression
structfun.1	⊢ 𝐹 Struct 𝑋

Assertion

Ref	Expression
structfun	⊢ Fun ◡◡𝐹

Proof of Theorem structfun

Step	Hyp	Ref	Expression
1		structfun.1	. 2 ⊢ 𝐹 Struct 𝑋
2		structfung 12492	. 2 ⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹)
3	1, 2	ax-mp 5	1 ⊢ Fun ◡◡𝐹

Syntax hints:   class class class wbr 4015  ^◡ccnv 4637  Fun wfun 5222   Struct cstr 12471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-struct 12477

This theorem is referenced by:  structfn  12494  strslfv  12520