Theorem fndmin 7054

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		dffn5 6957	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
2	1	biimpi 215	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3		df-mpt 5232	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
4	2, 3	eqtrdi 2784	. . . . 5 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
5		dffn5 6957	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
6	5	biimpi 215	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
7		df-mpt 5232	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}
8	6, 7	eqtrdi 2784	. . . . 5 ⊢ (𝐺 Fn 𝐴 → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))})
9	4, 8	ineqan12d 4214	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))}))
10		inopab 5831	. . . 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)))}
11	9, 10	eqtrdi 2784	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ∩ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
12	11	dmeqd 5908	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))})
13		19.42v 1950	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))))
14		anandi 675	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
15	14	exbii 1843	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))))
16		fvex 6910	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
17		eqeq1 2732	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 = (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
18	16, 17	ceqsexv 3523	. . . . . 6 ⊢ (∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥)) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))
19	18	anbi2i 622	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = (𝐹‘𝑥) ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))
20	13, 15, 19	3bitr3i 301	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥))) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))
21	20	abbii 2798	. . 3 ⊢ {𝑥 ∣ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
22		dmopab 5918	. . 3 ⊢ 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)))}
23		df-rab 3430	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))}
24	21, 22, 23	3eqtr4i 2766	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐺‘𝑥)))} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}
25	12, 24	eqtrdi 2784	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2705  {crab 3429   ∩ cin 3946  {copab 5210   ↦ cmpt 5231  dom cdm 5678   Fn wfn 6543  ‘cfv 6548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556

This theorem is referenced by:  fneqeql  7055  fninfp  7183  mgmhmeql  18676  mhmeql  18778  ghmeql  19193  lmhmeql  20940  hauseqlcld  23563  cvmliftmolem1  34891  cvmliftmolem2  34892  hausgraph  42633