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Theorem fvco2 5382
 Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 9-Oct-2004.) (Revised by set.mm contributors, 22-Oct-2011.)
Assertion
Ref Expression
fvco2 ((G Fn A C A) → ((F G) ‘C) = (F ‘(GC)))

Proof of Theorem fvco2
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnsnfv 5373 . . . . . . 7 ((G Fn A C A) → {(GC)} = (G “ {C}))
21imaeq2d 4942 . . . . . 6 ((G Fn A C A) → (F “ {(GC)}) = (F “ (G “ {C})))
3 imaco 5086 . . . . . 6 ((F G) “ {C}) = (F “ (G “ {C}))
42, 3syl6reqr 2404 . . . . 5 ((G Fn A C A) → ((F G) “ {C}) = (F “ {(GC)}))
54eqeq1d 2361 . . . 4 ((G Fn A C A) → (((F G) “ {C}) = {y} ↔ (F “ {(GC)}) = {y}))
65abbidv 2467 . . 3 ((G Fn A C A) → {y ((F G) “ {C}) = {y}} = {y (F “ {(GC)}) = {y}})
76unieqd 3902 . 2 ((G Fn A C A) → {y ((F G) “ {C}) = {y}} = {y (F “ {(GC)}) = {y}})
8 df-iota 4339 . . 3 (℩zC(F G)z) = {y {z C(F G)z} = {y}}
9 df-fv 4795 . . 3 ((F G) ‘C) = (℩zC(F G)z)
10 imasn 5018 . . . . . 6 ((F G) “ {C}) = {z C(F G)z}
1110eqeq1i 2360 . . . . 5 (((F G) “ {C}) = {y} ↔ {z C(F G)z} = {y})
1211abbii 2465 . . . 4 {y ((F G) “ {C}) = {y}} = {y {z C(F G)z} = {y}}
1312unieqi 3901 . . 3 {y ((F G) “ {C}) = {y}} = {y {z C(F G)z} = {y}}
148, 9, 133eqtr4i 2383 . 2 ((F G) ‘C) = {y ((F G) “ {C}) = {y}}
15 df-iota 4339 . . 3 (℩z(GC)Fz) = {y {z (GC)Fz} = {y}}
16 df-fv 4795 . . 3 (F ‘(GC)) = (℩z(GC)Fz)
17 imasn 5018 . . . . . 6 (F “ {(GC)}) = {z (GC)Fz}
1817eqeq1i 2360 . . . . 5 ((F “ {(GC)}) = {y} ↔ {z (GC)Fz} = {y})
1918abbii 2465 . . . 4 {y (F “ {(GC)}) = {y}} = {y {z (GC)Fz} = {y}}
2019unieqi 3901 . . 3 {y (F “ {(GC)}) = {y}} = {y {z (GC)Fz} = {y}}
2115, 16, 203eqtr4i 2383 . 2 (F ‘(GC)) = {y (F “ {(GC)}) = {y}}
227, 14, 213eqtr4g 2410 1 ((G Fn A C A) → ((F G) ‘C) = (F ‘(GC)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {csn 3737  ∪cuni 3891  ℩cio 4337   class class class wbr 4639   ∘ ccom 4721   “ cima 4722   Fn wfn 4776   ‘cfv 4781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  fvco  5383  fvco3  5384
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