Theorem dmfco 5597

Description: Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Assertion

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

Proof of Theorem dmfco

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

Step	Hyp	Ref	Expression
1		funfvex 5544	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐺‘𝐴) ∈ V)
2		opeq1 3790	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺‘𝐴), 𝑦⟩)
3	2	eleq1d 2256	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
4	3	ceqsexgv 2878	. . . . 5 ⊢ ((𝐺‘𝐴) ∈ V → (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
5	1, 4	syl 14	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
6		eqcom 2189	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝐴) ↔ (𝐺‘𝐴) = 𝑥)
7		funopfvb 5572	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
8	6, 7	bitrid 192	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺‘𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
9	8	anbi1d 465	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
10	9	exbidv 1835	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
11	5, 10	bitr3d 190	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
12	11	exbidv 1835	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13		eldm2g 4835	. . 3 ⊢ ((𝐺‘𝐴) ∈ V → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
14	1, 13	syl 14	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
15		eldm2g 4835	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
16		vex 2752	. . . . . 6 ⊢ 𝑦 ∈ V
17		opelco2g 4807	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
18	16, 17	mpan2 425	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
19	18	exbidv 1835	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	15, 19	bitrd 188	. . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
21	20	adantl 277	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
22	12, 14, 21	3bitr4rd 221	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363  ∃wex 1502   ∈ wcel 2158  Vcvv 2749  ⟨cop 3607  dom cdm 4638   ∘ ccom 4642  Fun wfun 5222  ‘cfv 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236

This theorem is referenced by:  ctssdccl  7124