Theorem dmfco 2864

Description: Domains of a function composition.

Assertion

Ref	Expression
dmfco	⊢ ((Fun G ∧ A ∈ dom G) → (A ∈ dom (F ∘ G) ↔ (G ‘A) ∈ dom F))

Proof of Theorem dmfco

Step	Hyp	Ref	Expression
1		eldmg 2529	. . . 4 ⊢ (A ∈ dom G → (A ∈ dom (F ∘ G) ↔ ∃y⟨A, y&ran3; ∈ (F ∘ G)))
2		visset 1350	. . . . . 6 ⊢ y ∈ V
3		opelcog 2511	. . . . . 6 ⊢ ((A ∈ dom G ∧ y ∈ V) → (⟨A, y⟩ ∈ (F ∘ G) ↔ ∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
4	2, 3	mpan2 519	. . . . 5 ⊢ (A ∈ dom G → (⟨A, y⟩ ∈ (F ∘ G) ↔ ∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
5	4	biexdv 936	. . . 4 ⊢ (A ∈ dom G → (∃y⟨A, y⟩ ∈ (F ∘ G) ↔ ∃y∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
6	1, 5	bitrd 406	. . 3 ⊢ (A ∈ dom G → (A ∈ dom (F ∘ G) ↔ ∃y∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
7	6	adantl 305	. 2 ⊢ ((Fun G ∧ A ∈ dom G) → (A ∈ dom (F ∘ G) ↔ ∃y∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
8		visset 1350	. . . . . . . . 9 ⊢ z ∈ V
9	8	funfvop 2857	. . . . . . . 8 ⊢ ((Fun G ∧ A ∈ dom G) → ((G ‘A) = z ↔ ⟨A, z⟩ ∈ G))
10		cleqcom 1103	. . . . . . . 8 ⊢ (z = (G ‘A) ↔ (G ‘A) = z)
11	9, 10	syl5bb 410	. . . . . . 7 ⊢ ((Fun G ∧ A ∈ dom G) → (z = (G ‘A) ↔ ⟨A, z⟩ ∈ G))
12	11	anbi1d 469	. . . . . 6 ⊢ ((Fun G ∧ A ∈ dom G) → ((z = (G ‘A) ∧ ⟨z, y⟩ ∈ F) ↔ (⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
13	12	biexdv 936	. . . . 5 ⊢ ((Fun G ∧ A ∈ dom G) → (∃z(z = (G ‘A) ∧ ⟨z, y⟩ ∈ F) ↔ ∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
14		fvex 2838	. . . . . 6 ⊢ (G ‘A) ∈ V
15		opeq1 1876	. . . . . . 7 ⊢ (z = (G ‘A) → ⟨z, y⟩ = ⟨(G ‘A), y⟩)
16	15	eleq1d 1155	. . . . . 6 ⊢ (z = (G ‘A) → (⟨z, y⟩ ∈ F ↔ ⟨(G ‘A), y⟩ ∈ F))
17	14, 16	ceqsexv 1371	. . . . 5 ⊢ (∃z(z = (G ‘A) ∧ ⟨z, y⟩ ∈ F) ↔ ⟨(G ‘A), y⟩ ∈ F)
18	13, 17	syl5bbr 412	. . . 4 ⊢ ((Fun G ∧ A ∈ dom G) → (⟨(G ‘A), y⟩ ∈ F ↔ ∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
19	18	biexdv 936	. . 3 ⊢ ((Fun G ∧ A ∈ dom G) → (∃y⟨(G ‘A), y⟩ ∈ F ↔ ∃y∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
20	14	eldm2 2528	. . 3 ⊢ ((G ‘A) ∈ dom F ↔ ∃y⟨(G ‘A), y⟩ ∈ F)
21	19, 20	syl5bb 410	. 2 ⊢ ((Fun G ∧ A ∈ dom G) → ((G ‘A) ∈ dom F ↔ ∃y∃z(⟨A, z⟩ ∈ G ∧ ⟨z, y⟩ ∈ F)))
22	7, 21	bitr4d 409	1 ⊢ ((Fun G ∧ A ∈ dom G) → (A ∈ dom (F ∘ G) ↔ (G ‘A) ∈ dom F))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  dom cdm 2410   ∘ ccom 2414  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  fvco 2865

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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