Theorem mpteqb 6299

Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6311. (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 3212	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2	1	ralimi 2952	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
3		fneq1 5979	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
4		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6019	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
7	6	mptfng 6019	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
8	3, 5, 7	3bitr4g 303	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
9	8	biimpd 219	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
10		r19.26 3064	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V))
11		nfmpt1 4747	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
12		nfmpt1 4747	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
13	11, 12	nfeq 2776	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
14		simpll 790	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fveq1d 6193	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
16	4	fvmpt2 6291	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
17	16	ad2ant2lr 784	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
18	6	fvmpt2 6291	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ V) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
19	18	ad2ant2l 782	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
20	15, 17, 19	3eqtr3d 2664	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ∧ 𝑥 ∈ 𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
21	20	exp31 630	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → (𝑥 ∈ 𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
22	13, 21	ralrimi 2957	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ∀𝑥 ∈ 𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23		ralim 2948	. . . . . . . 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	syl5bir 233	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) → ((∀𝑥 ∈ 𝐴 𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V) → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
26	25	expd 452	. . . . 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 2622	. . . 4 ⊢ 𝐴 = 𝐴
30		mpteq12 4736	. . . 4 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
31	29, 30	mpan 706	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
32	28, 31	impbid1 215	. 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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  eqfnfv  6311  eufnfv  6491  offveqb  6919  ramcl  15733  fucsect  16632  setcepi  16738  0frgp  18192  dprdf11  18422  dpjeq  18458  mvrf1  19425  mplmonmul  19464  frgpcyg  19922  ustuqtop  22050  mdegle0  23837  ply1nzb  23882  cvmliftphtlem  31299  matunitlindflem1  33405