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Theorem fnmptfvd 6807
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 3187 . . . 4 (𝜑 → ∀𝑎𝐴 𝐶𝑉)
4 eqid 2826 . . . . 5 (𝑎𝐴𝐶) = (𝑎𝐴𝐶)
54fnmpt 6485 . . . 4 (∀𝑎𝐴 𝐶𝑉 → (𝑎𝐴𝐶) Fn 𝐴)
63, 5syl 17 . . 3 (𝜑 → (𝑎𝐴𝐶) Fn 𝐴)
7 eqfnfv 6798 . . 3 ((𝑀 Fn 𝐴 ∧ (𝑎𝐴𝐶) Fn 𝐴) → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
81, 6, 7syl2anc 584 . 2 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖)))
9 fnmptfvd.s . . . . . . . 8 (𝑖 = 𝑎𝐷 = 𝐶)
109cbvmptv 5166 . . . . . . 7 (𝑖𝐴𝐷) = (𝑎𝐴𝐶)
1110eqcomi 2835 . . . . . 6 (𝑎𝐴𝐶) = (𝑖𝐴𝐷)
1211a1i 11 . . . . 5 (𝜑 → (𝑎𝐴𝐶) = (𝑖𝐴𝐷))
1312fveq1d 6669 . . . 4 (𝜑 → ((𝑎𝐴𝐶)‘𝑖) = ((𝑖𝐴𝐷)‘𝑖))
1413eqeq2d 2837 . . 3 (𝜑 → ((𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
1514ralbidv 3202 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑎𝐴𝐶)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖)))
16 simpr 485 . . . . 5 ((𝜑𝑖𝐴) → 𝑖𝐴)
17 fnmptfvd.d . . . . 5 ((𝜑𝑖𝐴) → 𝐷𝑈)
18 eqid 2826 . . . . . 6 (𝑖𝐴𝐷) = (𝑖𝐴𝐷)
1918fvmpt2 6775 . . . . 5 ((𝑖𝐴𝐷𝑈) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2016, 17, 19syl2anc 584 . . . 4 ((𝜑𝑖𝐴) → ((𝑖𝐴𝐷)‘𝑖) = 𝐷)
2120eqeq2d 2837 . . 3 ((𝜑𝑖𝐴) → ((𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ (𝑀𝑖) = 𝐷))
2221ralbidva 3201 . 2 (𝜑 → (∀𝑖𝐴 (𝑀𝑖) = ((𝑖𝐴𝐷)‘𝑖) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
238, 15, 223bitrd 306 1 (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  cmpt 5143   Fn wfn 6347  cfv 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360
This theorem is referenced by:  cramerlem1  21215  dssmapnvod  40234
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