Theorem mpteqb 6954

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6970. (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 3458	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 3070	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		fneq1 6577	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
4		eqid 2733	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6625	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		eqid 2733	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
7	6	mptfng 6625	. . . . . . 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 229	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
10		r19.26 3093	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
11		nfmpt1 5192	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfmpt1 5192	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
13	11, 12	nfeq 2909	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
14		simpll 766	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fveq1d 6830	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
16	4	fvmpt2 6946	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	ad2ant2lr 748	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
18	6	fvmpt2 6946	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
19	18	ad2ant2l 746	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
20	15, 17, 19	3eqtr3d 2776	. . . . . . . . . 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 3231	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23		ralim 3073	. . . . . . . 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 243	. . . . . 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 2733	. . . 4 ⊢ 𝐴 = 𝐴
30		mpteq12 5181	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
31	29, 30	mpan 690	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
32	28, 31	impbid1 225	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ↦ cmpt 5174   Fn wfn 6481  ‘cfv 6486

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  eqfnfv  6970  eufnfv  7169  offveqb  7643  caofidlcan  7654  ramcl  16943  fucsect  17884  setcepi  17997  0frgp  19693  dprdf11  19939  dpjeq  19975  frgpcyg  21512  mvrf1  21924  mplmonmul  21972  ustuqtop  24162  mdegle0  26010  ply1nzb  26056  fedgmullem2  33664  cvmliftphtlem  35382  matunitlindflem1  37677  cfsetsnfsetf1  47184  1arymaptf1  48768