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 ‘(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 5373	. . . . . . 7 ⊢ ((G Fn A ∧ C ∈ A) → {(G ‘C)} = (G “ {C}))
2	1	imaeq2d 4942	. . . . . 6 ⊢ ((G Fn A ∧ C ∈ A) → (F “ {(G ‘C)}) = (F “ (G “ {C})))
3		imaco 5086	. . . . . 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 2467	. . 3 ⊢ ((G Fn A ∧ C ∈ A) → {y ∣ ((F ∘ G) “ {C}) = {y}} = {y ∣ (F “ {(G ‘C)}) = {y}})
7	6	unieqd 3902	. 2 ⊢ ((G Fn A ∧ C ∈ A) → ∪{y ∣ ((F ∘ G) “ {C}) = {y}} = ∪{y ∣ (F “ {(G ‘C)}) = {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}
11	10	eqeq1i 2360	. . . . 5 ⊢ (((F ∘ G) “ {C}) = {y} ↔ {z ∣ C(F ∘ G)z} = {y})
12	11	abbii 2465	. . . 4 ⊢ {y ∣ ((F ∘ G) “ {C}) = {y}} = {y ∣ {z ∣ C(F ∘ G)z} = {y}}
13	12	unieqi 3901	. . 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 4339	. . 3 ⊢ (℩z(G ‘C)Fz) = ∪{y ∣ {z ∣ (G ‘C)Fz} = {y}}
16		df-fv 4795	. . 3 ⊢ (F ‘(G ‘C)) = (℩z(G ‘C)Fz)
17		imasn 5018	. . . . . 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 2465	. . . 4 ⊢ {y ∣ (F “ {(G ‘C)}) = {y}} = {y ∣ {z ∣ (G ‘C)Fz} = {y}}
20	19	unieqi 3901	. . 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 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