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Theorem nat1st2nd 16812
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 16723 . . . 4 Rel (𝐶 Func 𝐷)
3 natrcl.1 . . . . . . 7 𝑁 = (𝐶 Nat 𝐷)
43natrcl 16811 . . . . . 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 477 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 1st2nd 7381 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
82, 6, 7sylancr 698 . . 3 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
95simprd 482 . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
10 1st2nd 7381 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
112, 9, 10sylancr 698 . . 3 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
128, 11oveq12d 6831 . 2 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
131, 12eleqtrd 2841 1 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cop 4327  Rel wrel 5271  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332   Func cfunc 16715   Nat cnat 16802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ixp 8075  df-func 16719  df-nat 16804
This theorem is referenced by:  fuccocl  16825  fuclid  16827  fucrid  16828  fucass  16829  fucsect  16833  invfuc  16835  fucpropd  16838  evlfcllem  17062  evlfcl  17063  curfuncf  17079  yonedalem3a  17115  yonedalem3b  17120  yonedainv  17122  yonffthlem  17123
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