Theorem fndmin 5603

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 5542	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2		df-mpt 4052	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
3	1, 2	eqtrdi 2219	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
4		dffn5im 5542	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
5		df-mpt 4052	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
6	4, 5	eqtrdi 2219	. . . . 5 ⊢ (𝐺 Fn 𝐴 → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))})
7	3, 6	ineqan12d 3330	. . . 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 4743	. . . 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 2219	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
10	9	dmeqd 4813	. 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 585	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
12	11	exbii 1598	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
13		19.42v 1899	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
14	12, 13	bitr3i 185	. . . . . 6 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
15		funfvex 5513	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
16		eqeq1 2177	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 = (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
17	16	ceqsexgv 2859	. . . . . . . . 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 5298	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
20	19	pm5.32da 449	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
21	14, 20	syl5bb 191	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
22	21	abbidv 2288	. . . 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 4822	. . . 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 2457	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
25	22, 23, 24	3eqtr4g 2228	. . 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 274	. 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 2203	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  {crab 2452  Vcvv 2730   ∩ cin 3120  {copab 4049   ↦ cmpt 4050  dom cdm 4611  Fun wfun 5192   Fn wfn 5193  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  fneqeql  5604  mhmeql  12707