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Theorem elhomai2 16891
Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h 𝐻 = (Homa𝐶)
homafval.b 𝐵 = (Base‘𝐶)
homafval.c (𝜑𝐶 ∈ Cat)
homaval.j 𝐽 = (Hom ‘𝐶)
homaval.x (𝜑𝑋𝐵)
homaval.y (𝜑𝑌𝐵)
elhomai.f (𝜑𝐹 ∈ (𝑋𝐽𝑌))
Assertion
Ref Expression
elhomai2 (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Proof of Theorem elhomai2
StepHypRef Expression
1 df-ot 4326 . 2 𝑋, 𝑌, 𝐹⟩ = ⟨⟨𝑋, 𝑌⟩, 𝐹
2 homarcl.h . . . 4 𝐻 = (Homa𝐶)
3 homafval.b . . . 4 𝐵 = (Base‘𝐶)
4 homafval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 homaval.j . . . 4 𝐽 = (Hom ‘𝐶)
6 homaval.x . . . 4 (𝜑𝑋𝐵)
7 homaval.y . . . 4 (𝜑𝑌𝐵)
8 elhomai.f . . . 4 (𝜑𝐹 ∈ (𝑋𝐽𝑌))
92, 3, 4, 5, 6, 7, 8elhomai 16890 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
10 df-br 4788 . . 3 (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
119, 10sylib 208 . 2 (𝜑 → ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
121, 11syl5eqel 2854 1 (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cop 4323  cotp 4325   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  Hom chom 16160  Catccat 16532  Homachoma 16880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-ot 4326  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-homa 16883
This theorem is referenced by:  idahom  16917  coahom  16927
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