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Theorem elhomai2 17289
 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 4537 . 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 17288 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
10 df-br 5034 . . 3 (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
119, 10sylib 221 . 2 (𝜑 → ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
121, 11eqeltrid 2897 1 (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  ⟨cop 4534  ⟨cotp 4536   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  Hom chom 16571  Catccat 16930  Homachoma 17278 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-ot 4537  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-homa 17281 This theorem is referenced by:  idahom  17315  coahom  17325
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