Theorem fnmptfvd 7060

Description: A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)

Hypotheses

Ref	Expression
fnmptfvd.m	⊢ (𝜑 → 𝑀 Fn 𝐴)
fnmptfvd.s	⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶)
fnmptfvd.d	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)
fnmptfvd.c	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fnmptfvd	⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))

Distinct variable groups:   𝐴,𝑎,𝑖   𝐶,𝑖   𝐷,𝑎   𝑀,𝑎,𝑖   𝑈,𝑎,𝑖   𝑉,𝑎,𝑖   𝜑,𝑎,𝑖

Allowed substitution hints:   𝐶(𝑎)   𝐷(𝑖)

Proof of Theorem fnmptfvd

Step	Hyp	Ref	Expression
1		fnmptfvd.m	. . 3 ⊢ (𝜑 → 𝑀 Fn 𝐴)
2		fnmptfvd.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉)
3	2	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉)
4		eqid 2763	. . . . 5 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ 𝐶)
5	4	fnmpt 6698	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
6	3, 5	syl 18	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
7		eqfnfv 7049	. . 3 ⊢ ((𝑀 Fn 𝐴 ∧ (𝑎 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖)))
8	1, 6, 7	syl2anc 596	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖)))
9		fnmptfvd.s	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶)
10	9	cbvmptv 5212	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑎 ∈ 𝐴 ↦ 𝐶)
11	10	eqcomi 2772	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷)
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ 𝐶) = (𝑖 ∈ 𝐴 ↦ 𝐷))
13	12	fveq1d 6907	. . . 4 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖))
14	13	eqeq2d 2774	. . 3 ⊢ (𝜑 → ((𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)))
15	14	ralbidv 3189	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑎 ∈ 𝐴 ↦ 𝐶)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖)))
16		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴)
17		fnmptfvd.d	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)
18		eqid 2763	. . . . . 6 ⊢ (𝑖 ∈ 𝐴 ↦ 𝐷) = (𝑖 ∈ 𝐴 ↦ 𝐷)
19	18	fvmpt2 7025	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝐷 ∈ 𝑈) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷)
20	16, 17, 19	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) = 𝐷)
21	20	eqeq2d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ (𝑀‘𝑖) = 𝐷))
22	21	ralbidva 3187	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = ((𝑖 ∈ 𝐴 ↦ 𝐷)‘𝑖) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))
23	8, 15, 22	3bitrd 308	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ↦ cmpt 5189   Fn wfn 6550  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-fv 6563

This theorem is used by:  cramerlem1  23032  dssmapnvod  45210