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Theorem nat1st2nd 17217
Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1 𝑁 = (𝐶 Nat 𝐷)
nat1st2nd.2 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
Assertion
Ref Expression
nat1st2nd (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))

Proof of Theorem nat1st2nd
StepHypRef Expression
1 nat1st2nd.2 . 2 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
2 relfunc 17128 . . . 4 Rel (𝐶 Func 𝐷)
3 natrcl.1 . . . . . . 7 𝑁 = (𝐶 Nat 𝐷)
43natrcl 17216 . . . . . 6 (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
51, 4syl 17 . . . . 5 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
65simpld 498 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 1st2nd 7724 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
82, 6, 7sylancr 590 . . 3 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
95simprd 499 . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
10 1st2nd 7724 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
112, 9, 10sylancr 590 . . 3 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
128, 11oveq12d 7157 . 2 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
131, 12eleqtrd 2895 1 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cop 4534  Rel wrel 5528  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674   Func cfunc 17120   Nat cnat 17207
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-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-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-ixp 8449  df-func 17124  df-nat 17209
This theorem is referenced by:  fuccocl  17230  fuclid  17232  fucrid  17233  fucass  17234  fucsect  17238  invfuc  17240  fucpropd  17243  evlfcllem  17467  evlfcl  17468  curfuncf  17484  yonedalem3a  17520  yonedalem3b  17525  yonedainv  17527  yonffthlem  17528
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