Theorem mpteqb 7013

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 7029. (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 3471	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		fneq1 6630	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
4		eqid 2760	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6678	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		eqid 2760	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
7	6	mptfng 6678	. . . . . . 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 3122	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
11		nfmpt1 5203	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfmpt1 5203	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
13	11, 12	nfeq 2935	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
14		simpll 779	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fveq1d 6887	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
16	4	fvmpt2 7005	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	ad2ant2lr 761	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
18	6	fvmpt2 7005	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
19	18	ad2ant2l 759	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
20	15, 17, 19	3eqtr3d 2803	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
21	20	exp31 425	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑥 ∈ 𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
22	13, 21	ralrimi 3260	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23		ralim 3102	. . . . . . . 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 421	. . . . 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 2760	. . . 4 ⊢ 𝐴 = 𝐴
30		mpteq12 5192	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
31	29, 30	mpan 703	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ↦ cmpt 5185   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  eqfnfv  7029  eufnfv  7235  offveqb  7720  caofidlcan  7731  ramcl  17212  fucsect  18155  setcepi  18268  0frgp  19998  dprdf11  20244  dpjeq  20280  frgpcyg  21884  mvrf1  22298  mplmonmul  22350  matunitlindflem1  22999  ustuqtop  24570  mdegle0  26400  ply1nzb  26446  psrmonmul  34228  fedgmullem2  34308  cvmliftphtlem  36168  cfsetsnfsetf1  48222  1arymaptf1  49847