Theorem mpteqb 7017

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 7032. (Contributed by Mario Carneiro, 14-Nov-2014.)

Assertion

Ref	Expression
mpteqb	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem mpteqb

Step	Hyp	Ref	Expression
1		elex 3492	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 3082	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		fneq1 6640	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
4		eqid 2731	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6689	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		eqid 2731	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
7	6	mptfng 6689	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
8	3, 5, 7	3bitr4g 314	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
9	8	biimpd 228	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
10		r19.26 3110	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
11		nfmpt1 5256	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfmpt1 5256	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
13	11, 12	nfeq 2915	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
14		simpll 764	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fveq1d 6893	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
16	4	fvmpt2 7009	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	ad2ant2lr 745	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
18	6	fvmpt2 7009	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
19	18	ad2ant2l 743	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
20	15, 17, 19	3eqtr3d 2779	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
21	20	exp31 419	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑥 ∈ 𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
22	13, 21	ralrimi 3253	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23		ralim 3085	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶) → (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
25	10, 24	biimtrrid 242	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
26	25	expd 415	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝐶 ∈ V → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)))
27	9, 26	mpdd 43	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
28	27	com12 32	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
29		eqid 2731	. . . 4 ⊢ 𝐴 = 𝐴
30		mpteq12 5240	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
31	29, 30	mpan 687	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
32	28, 31	impbid1 224	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
33	2, 32	syl 17	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ↦ cmpt 5231   Fn wfn 6538  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  eqfnfv  7032  eufnfv  7233  offveqb  7699  ramcl  16969  fucsect  17935  setcepi  18048  0frgp  19695  dprdf11  19941  dpjeq  19977  frgpcyg  21439  mvrf1  21856  mplmonmul  21902  ustuqtop  24071  mdegle0  25933  ply1nzb  25978  fedgmullem2  33169  cvmliftphtlem  34772  matunitlindflem1  36948  cfsetsnfsetf1  46228  1arymaptf1  47490