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Theorem mpteqb 7034
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 7050. (Contributed by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
mpteqb (∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem mpteqb
StepHypRef Expression
1 elex 3498 . . 3 (𝐵𝑉𝐵 ∈ V)
21ralimi 3080 . 2 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 fneq1 6659 . . . . . . 7 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((𝑥𝐴𝐵) Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
4 eqid 2734 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 6707 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
6 eqid 2734 . . . . . . . 8 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
76mptfng 6707 . . . . . . 7 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
83, 5, 73bitr4g 314 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 𝐶 ∈ V))
98biimpd 229 . . . . 5 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → ∀𝑥𝐴 𝐶 ∈ V))
10 r19.26 3108 . . . . . . 7 (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V))
11 nfmpt1 5255 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
12 nfmpt1 5255 . . . . . . . . . 10 𝑥(𝑥𝐴𝐶)
1311, 12nfeq 2916 . . . . . . . . 9 𝑥(𝑥𝐴𝐵) = (𝑥𝐴𝐶)
14 simpll 767 . . . . . . . . . . . 12 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
1514fveq1d 6908 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = ((𝑥𝐴𝐶)‘𝑥))
164fvmpt2 7026 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ V) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716ad2ant2lr 748 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
186fvmpt2 7026 . . . . . . . . . . . 12 ((𝑥𝐴𝐶 ∈ V) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1918ad2ant2l 746 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
2015, 17, 193eqtr3d 2782 . . . . . . . . . 10 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
2120exp31 419 . . . . . . . . 9 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (𝑥𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
2213, 21ralrimi 3254 . . . . . . . 8 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23 ralim 3083 . . . . . . . 8 (∀𝑥𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶) → (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2422, 23syl 17 . . . . . . 7 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2510, 24biimtrrid 243 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2625expd 415 . . . . 5 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → (∀𝑥𝐴 𝐶 ∈ V → ∀𝑥𝐴 𝐵 = 𝐶)))
279, 26mpdd 43 . . . 4 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → ∀𝑥𝐴 𝐵 = 𝐶))
2827com12 32 . . 3 (∀𝑥𝐴 𝐵 ∈ V → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 𝐵 = 𝐶))
29 eqid 2734 . . . 4 𝐴 = 𝐴
30 mpteq12 5239 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3129, 30mpan 690 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3228, 31impbid1 225 . 2 (∀𝑥𝐴 𝐵 ∈ V → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
332, 32syl 17 1 (∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cmpt 5230   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  eqfnfv  7050  eufnfv  7248  offveqb  7723  ramcl  17062  fucsect  18028  setcepi  18141  0frgp  19811  dprdf11  20057  dpjeq  20093  frgpcyg  21609  mvrf1  22023  mplmonmul  22071  ustuqtop  24270  mdegle0  26130  ply1nzb  26176  fedgmullem2  33657  cvmliftphtlem  35301  matunitlindflem1  37602  cfsetsnfsetf1  47008  1arymaptf1  48491
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