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Theorem mpteqb 6441
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6454. (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 3364 . . 3 (𝐵𝑉𝐵 ∈ V)
21ralimi 3101 . 2 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 fneq1 6119 . . . . . . 7 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((𝑥𝐴𝐵) Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
4 eqid 2771 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 6159 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
6 eqid 2771 . . . . . . . 8 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
76mptfng 6159 . . . . . . 7 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
83, 5, 73bitr4g 303 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 𝐶 ∈ V))
98biimpd 219 . . . . 5 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → ∀𝑥𝐴 𝐶 ∈ V))
10 r19.26 3212 . . . . . . 7 (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V))
11 nfmpt1 4881 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
12 nfmpt1 4881 . . . . . . . . . 10 𝑥(𝑥𝐴𝐶)
1311, 12nfeq 2925 . . . . . . . . 9 𝑥(𝑥𝐴𝐵) = (𝑥𝐴𝐶)
14 simpll 750 . . . . . . . . . . . 12 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
1514fveq1d 6334 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = ((𝑥𝐴𝐶)‘𝑥))
164fvmpt2 6433 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ V) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716ad2ant2lr 742 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
186fvmpt2 6433 . . . . . . . . . . . 12 ((𝑥𝐴𝐶 ∈ V) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1918ad2ant2l 740 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
2015, 17, 193eqtr3d 2813 . . . . . . . . . 10 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
2120exp31 406 . . . . . . . . 9 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (𝑥𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
2213, 21ralrimi 3106 . . . . . . . 8 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23 ralim 3097 . . . . . . . 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, 24syl5bir 233 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2625expd 400 . . . . 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 2771 . . . 4 𝐴 = 𝐴
30 mpteq12 4870 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3129, 30mpan 670 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3228, 31impbid1 215 . 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cmpt 4863   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  eqfnfv  6454  eufnfv  6634  offveqb  7066  ramcl  15940  fucsect  16839  setcepi  16945  0frgp  18399  dprdf11  18630  dpjeq  18666  mvrf1  19640  mplmonmul  19679  frgpcyg  20137  ustuqtop  22270  mdegle0  24057  ply1nzb  24102  cvmliftphtlem  31637  matunitlindflem1  33738
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