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Theorem 2ndf2 17148
Description: Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfval.t 𝑇 = (𝐶 ×c 𝐷)
1stfval.b 𝐵 = (Base‘𝑇)
1stfval.h 𝐻 = (Hom ‘𝑇)
1stfval.c (𝜑𝐶 ∈ Cat)
1stfval.d (𝜑𝐷 ∈ Cat)
2ndfval.p 𝑄 = (𝐶 2ndF 𝐷)
2ndf1.p (𝜑𝑅𝐵)
2ndf2.p (𝜑𝑆𝐵)
Assertion
Ref Expression
2ndf2 (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

Proof of Theorem 2ndf2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 1stfval.b . . . 4 𝐵 = (Base‘𝑇)
3 1stfval.h . . . 4 𝐻 = (Hom ‘𝑇)
4 1stfval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 1stfval.d . . . 4 (𝜑𝐷 ∈ Cat)
6 2ndfval.p . . . 4 𝑄 = (𝐶 2ndF 𝐷)
71, 2, 3, 4, 5, 62ndfval 17146 . . 3 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8 fo2nd 7421 . . . . . 6 2nd :V–onto→V
9 fofun 6331 . . . . . 6 (2nd :V–onto→V → Fun 2nd )
108, 9ax-mp 5 . . . . 5 Fun 2nd
112fvexi 6424 . . . . 5 𝐵 ∈ V
12 resfunexg 6707 . . . . 5 ((Fun 2nd𝐵 ∈ V) → (2nd𝐵) ∈ V)
1310, 11, 12mp2an 684 . . . 4 (2nd𝐵) ∈ V
1411, 11mpt2ex 7482 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
1513, 14op2ndd 7411 . . 3 (𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (2nd𝑄) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
167, 15syl 17 . 2 (𝜑 → (2nd𝑄) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
17 simprl 788 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑥 = 𝑅)
18 simprr 790 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑦 = 𝑆)
1917, 18oveq12d 6895 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2019reseq2d 5599 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (2nd ↾ (𝑥𝐻𝑦)) = (2nd ↾ (𝑅𝐻𝑆)))
21 2ndf1.p . 2 (𝜑𝑅𝐵)
22 2ndf2.p . 2 (𝜑𝑆𝐵)
23 ovex 6909 . . . 4 (𝑅𝐻𝑆) ∈ V
24 resfunexg 6707 . . . 4 ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
2510, 23, 24mp2an 684 . . 3 (2nd ↾ (𝑅𝐻𝑆)) ∈ V
2625a1i 11 . 2 (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
2716, 20, 21, 22, 26ovmpt2d 7021 1 (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3384  cop 4373  cres 5313  Fun wfun 6094  ontowfo 6098  cfv 6100  (class class class)co 6877  cmpt2 6879  2nd c2nd 7399  Basecbs 16181  Hom chom 16275  Catccat 16636   ×c cxpc 17120   2ndF c2ndf 17122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182  ax-cnex 10279  ax-resscn 10280  ax-1cn 10281  ax-icn 10282  ax-addcl 10283  ax-addrcl 10284  ax-mulcl 10285  ax-mulrcl 10286  ax-mulcom 10287  ax-addass 10288  ax-mulass 10289  ax-distr 10290  ax-i2m1 10291  ax-1ne0 10292  ax-1rid 10293  ax-rnegex 10294  ax-rrecex 10295  ax-cnre 10296  ax-pre-lttri 10297  ax-pre-lttrn 10298  ax-pre-ltadd 10299  ax-pre-mulgt0 10300
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-pss 3784  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4628  df-int 4667  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5897  df-ord 5943  df-on 5944  df-lim 5945  df-suc 5946  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7400  df-2nd 7401  df-wrecs 7644  df-recs 7706  df-rdg 7744  df-1o 7798  df-oadd 7802  df-er 7981  df-en 8195  df-dom 8196  df-sdom 8197  df-fin 8198  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10557  df-neg 10558  df-nn 11312  df-2 11373  df-3 11374  df-4 11375  df-5 11376  df-6 11377  df-7 11378  df-8 11379  df-9 11380  df-n0 11578  df-z 11664  df-dec 11781  df-uz 11928  df-fz 12578  df-struct 16183  df-ndx 16184  df-slot 16185  df-base 16187  df-hom 16288  df-cco 16289  df-xpc 17124  df-2ndf 17126
This theorem is referenced by:  2ndfcl  17150  prf2nd  17157  1st2ndprf  17158  uncf2  17189  curf2ndf  17199  yonedalem22  17230
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