Theorem fndmin 5666

Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
fndmin	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝐴

Proof of Theorem fndmin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5im 5603	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		df-mpt 4093	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
3	1, 2	eqtrdi 2242	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
4		dffn5im 5603	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
5		df-mpt 4093	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
6	4, 5	eqtrdi 2242	. . . . 5 ⊢ (𝐺 Fn 𝐴 → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))})
7	3, 6	ineqan12d 3363	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}))
8		inopab 4795	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))}
9	7, 8	eqtrdi 2242	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
10	9	dmeqd 4865	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
11		anandi 590	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
12	11	exbii 1616	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
13		19.42v 1918	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
14	12, 13	bitr3i 186	. . . . . 6 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
15		funfvex 5572	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
16		eqeq1 2200	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 = (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
17	16	ceqsexgv 2890	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ V → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
18	15, 17	syl 14	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
19	18	funfni 5355	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
20	19	pm5.32da 452	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
21	14, 20	bitrid 192	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
22	21	abbidv 2311	. . . 4 ⊢ (𝐹 Fn 𝐴 → {𝑥 ∣ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))})
23		dmopab 4874	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∣ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))}
24		df-rab 2481	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
25	22, 23, 24	3eqtr4g 2251	. . 3 ⊢ (𝐹 Fn 𝐴 → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
26	25	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
27	10, 26	eqtrd 2226	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  {crab 2476  Vcvv 2760   ∩ cin 3153  {copab 4090   ↦ cmpt 4091  dom cdm 4660  Fun wfun 5249   Fn wfn 5250  ‘cfv 5255

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  fneqeql  5667  mhmeql  13067  ghmeql  13340