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Theorem fnmptfvd 6484
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 (𝜑𝑀 Fn 𝐴)
fnmptfvd.s (𝑖 = 𝑎𝐷 = 𝐶)
fnmptfvd.d ((𝜑𝑖𝐴) → 𝐷𝑈)
fnmptfvd.c ((𝜑𝑎𝐴) → 𝐶𝑉)
Assertion
Ref Expression
fnmptfvd (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 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
StepHypRef Expression
1 fnmptfvd.m . . 3 (𝜑𝑀 Fn 𝐴)
2 fnmptfvd.c . . . . 5 ((𝜑𝑎𝐴) → 𝐶𝑉)
32ralrimiva 3104 . . . 4 (𝜑 → ∀𝑎𝐴 𝐶𝑉)
4 eqid 2760 . . . . 5 (𝑎𝐴𝐶) = (𝑎𝐴𝐶)
54fnmpt 6181 . . . 4 (∀𝑎𝐴 𝐶𝑉 → (𝑎𝐴𝐶) Fn 𝐴)
63, 5syl 17 . . 3 (𝜑 → (𝑎𝐴𝐶) Fn 𝐴)
7 eqfnfv 6475 . . 3 ((𝑀 Fn 𝐴 ∧ (𝑎𝐴𝐶) Fn 𝐴) → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
81, 6, 7syl2anc 696 . 2 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
9 fnmptfvd.s . . . . . . . 8 (𝑖 = 𝑎𝐷 = 𝐶)
109cbvmptv 4902 . . . . . . 7 (𝑖𝐴𝐷) = (𝑎𝐴𝐶)
1110eqcomi 2769 . . . . . 6 (𝑎𝐴𝐶) = (𝑖𝐴𝐷)
1211a1i 11 . . . . 5 (𝜑 → (𝑎𝐴𝐶) = (𝑖𝐴𝐷))
1312fveq1d 6355 . . . 4 (𝜑 → ((𝑎𝐴𝐶)‘𝑖) = ((𝑖𝐴𝐷)‘𝑖))
1413eqeq2d 2770 . . 3 (𝜑 → ((𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
1514ralbidv 3124 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
16 simpr 479 . . . . 5 ((𝜑𝑖𝐴) → 𝑖𝐴)
17 fnmptfvd.d . . . . 5 ((𝜑𝑖𝐴) → 𝐷𝑈)
18 eqid 2760 . . . . . 6 (𝑖𝐴𝐷) = (𝑖𝐴𝐷)
1918fvmpt2 6454 . . . . 5 ((𝑖𝐴𝐷𝑈) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2016, 17, 19syl2anc 696 . . . 4 ((𝜑𝑖𝐴) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2120eqeq2d 2770 . . 3 ((𝜑𝑖𝐴) → ((𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ (𝑀𝑖) = 𝐷))
2221ralbidva 3123 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
238, 15, 223bitrd 294 1 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cmpt 4881   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  cramerlem1  20715  dssmapnvod  38834
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