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Theorem nat1st2nd 17969
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 17876 . . . 4 Rel (𝐶 Func 𝐷)
3 natrcl.1 . . . . . . 7 𝑁 = (𝐶 Nat 𝐷)
43natrcl 17968 . . . . . 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 493 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 1st2nd 8045 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
82, 6, 7sylancr 585 . . 3 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
95simprd 494 . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
10 1st2nd 8045 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
112, 9, 10sylancr 585 . . 3 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
128, 11oveq12d 7434 . 2 (𝜑 → (𝐹𝑁𝐺) = (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
131, 12eleqtrd 2828 1 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cop 4629  Rel wrel 5679  cfv 6546  (class class class)co 7416  1st c1st 7993  2nd c2nd 7994   Func cfunc 17868   Nat cnat 17959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-ixp 8919  df-func 17872  df-nat 17961
This theorem is referenced by:  fuccocl  17984  fuclid  17986  fucrid  17987  fucass  17988  fucsect  17992  invfuc  17994  fucpropd  17997  evlfcllem  18241  evlfcl  18242  curfuncf  18258  yonedalem3a  18294  yonedalem3b  18299  yonedainv  18301  yonffthlem  18302
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