Theorem fvco2 5383

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 ‘(G ‘C)))

Proof of Theorem fvco2

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnsnfv 5374	. . . . . . 7 ⊢ ((G Fn A ∧ C ∈ A) → {(G ‘C)} = (G “ {C}))
2	1	imaeq2d 4943	. . . . . 6 ⊢ ((G Fn A ∧ C ∈ A) → (F “ {(G ‘C)}) = (F “ (G “ {C})))
3		imaco 5087	. . . . . 6 ⊢ ((F ∘ G) “ {C}) = (F “ (G “ {C}))
4	2, 3	syl6reqr 2404	. . . . 5 ⊢ ((G Fn A ∧ C ∈ A) → ((F ∘ G) “ {C}) = (F “ {(G ‘C)}))
5	4	eqeq1d 2361	. . . 4 ⊢ ((G Fn A ∧ C ∈ A) → (((F ∘ G) “ {C}) = {y} ↔ (F “ {(G ‘C)}) = {y}))
6	5	abbidv 2468	. . 3 ⊢ ((G Fn A ∧ C ∈ A) → {y ∣ ((F ∘ G) “ {C}) = {y}} = {y ∣ (F “ {(G ‘C)}) = {y}})
7	6	unieqd 3903	. 2 ⊢ ((G Fn A ∧ C ∈ A) → ∪{y ∣ ((F ∘ G) “ {C}) = {y}} = ∪{y ∣ (F “ {(G ‘C)}) = {y}})
8		df-iota 4340	. . 3 ⊢ (℩zC(F ∘ G)z) = ∪{y ∣ {z ∣ C(F ∘ G)z} = {y}}
9		df-fv 4796	. . 3 ⊢ ((F ∘ G) ‘C) = (℩zC(F ∘ G)z)
10		imasn 5019	. . . . . 6 ⊢ ((F ∘ G) “ {C}) = {z ∣ C(F ∘ G)z}
11	10	eqeq1i 2360	. . . . 5 ⊢ (((F ∘ G) “ {C}) = {y} ↔ {z ∣ C(F ∘ G)z} = {y})
12	11	abbii 2466	. . . 4 ⊢ {y ∣ ((F ∘ G) “ {C}) = {y}} = {y ∣ {z ∣ C(F ∘ G)z} = {y}}
13	12	unieqi 3902	. . 3 ⊢ ∪{y ∣ ((F ∘ G) “ {C}) = {y}} = ∪{y ∣ {z ∣ C(F ∘ G)z} = {y}}
14	8, 9, 13	3eqtr4i 2383	. 2 ⊢ ((F ∘ G) ‘C) = ∪{y ∣ ((F ∘ G) “ {C}) = {y}}
15		df-iota 4340	. . 3 ⊢ (℩z(G ‘C)Fz) = ∪{y ∣ {z ∣ (G ‘C)Fz} = {y}}
16		df-fv 4796	. . 3 ⊢ (F ‘(G ‘C)) = (℩z(G ‘C)Fz)
17		imasn 5019	. . . . . 6 ⊢ (F “ {(G ‘C)}) = {z ∣ (G ‘C)Fz}
18	17	eqeq1i 2360	. . . . 5 ⊢ ((F “ {(G ‘C)}) = {y} ↔ {z ∣ (G ‘C)Fz} = {y})
19	18	abbii 2466	. . . 4 ⊢ {y ∣ (F “ {(G ‘C)}) = {y}} = {y ∣ {z ∣ (G ‘C)Fz} = {y}}
20	19	unieqi 3902	. . 3 ⊢ ∪{y ∣ (F “ {(G ‘C)}) = {y}} = ∪{y ∣ {z ∣ (G ‘C)Fz} = {y}}
21	15, 16, 20	3eqtr4i 2383	. 2 ⊢ (F ‘(G ‘C)) = ∪{y ∣ (F “ {(G ‘C)}) = {y}}
22	7, 14, 21	3eqtr4g 2410	1 ⊢ ((G Fn A ∧ C ∈ A) → ((F ∘ G) ‘C) = (F ‘(G ‘C)))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {csn 3738  ∪cuni 3892  ℩cio 4338   class class class wbr 4640   ∘ ccom 4722   “ cima 4723   Fn wfn 4777   ‘cfv 4782

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  fvco  5384  fvco3  5385