Theorem mpteqb 7011

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 7027. (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 3476	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 3102	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		fneq1 6628	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
4		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6676	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
7	6	mptfng 6676	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
8	3, 5, 7	3bitr4g 317	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
9	8	biimpd 232	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
10		r19.26 3125	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
11		nfmpt1 5211	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfmpt1 5211	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
13	11, 12	nfeq 2938	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
14		simpll 778	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fveq1d 6885	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
16	4	fvmpt2 7003	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	ad2ant2lr 760	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
18	6	fvmpt2 7003	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
19	18	ad2ant2l 758	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
20	15, 17, 19	3eqtr3d 2806	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
21	20	exp31 424	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑥 ∈ 𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
22	13, 21	ralrimi 3263	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23		ralim 3105	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶) → (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
24	22, 23	syl 18	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
25	10, 24	biimtrrid 246	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
26	25	expd 420	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → (∀𝑥 ∈ 𝐴 𝐶 ∈ V → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)))
27	9, 26	mpdd 44	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
28	27	com12 33	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
29		eqid 2763	. . . 4 ⊢ 𝐴 = 𝐴
30		mpteq12 5200	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
31	29, 30	mpan 702	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
32	28, 31	impbid1 228	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
33	2, 32	syl 18	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ↦ cmpt 5193   Fn wfn 6533  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is referenced by:  eqfnfv  7027  eufnfv  7229  offveqb  7703  caofidlcan  7714  ramcl  17090  fucsect  18033  setcepi  18146  0frgp  19850  dprdf11  20096  dpjeq  20132  frgpcyg  21704  mvrf1  22116  mplmonmul  22168  ustuqtop  24384  mdegle0  26215  ply1nzb  26261  psrmonmul  33918  fedgmullem2  33998  cvmliftphtlem  35787  matunitlindflem1  38245  cfsetsnfsetf1  47773  1arymaptf1  49399