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Theorem mpteqb 6031
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6043. (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 3075 . . 3 (𝐵𝑉𝐵 ∈ V)
21ralimi 2831 . 2 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 fneq1 5719 . . . . . . 7 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((𝑥𝐴𝐵) Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
4 eqid 2505 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 5758 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
6 eqid 2505 . . . . . . . 8 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
76mptfng 5758 . . . . . . 7 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
83, 5, 73bitr4g 298 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 𝐶 ∈ V))
98biimpd 214 . . . . 5 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → ∀𝑥𝐴 𝐶 ∈ V))
10 r19.26 2939 . . . . . . 7 (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V))
11 nfmpt1 4525 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
12 nfmpt1 4525 . . . . . . . . . 10 𝑥(𝑥𝐴𝐶)
1311, 12nfeq 2657 . . . . . . . . 9 𝑥(𝑥𝐴𝐵) = (𝑥𝐴𝐶)
14 simpll 777 . . . . . . . . . . . 12 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
1514fveq1d 5929 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = ((𝑥𝐴𝐶)‘𝑥))
164fvmpt2 6024 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ V) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716ad2ant2lr 771 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
186fvmpt2 6024 . . . . . . . . . . . 12 ((𝑥𝐴𝐶 ∈ V) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1918ad2ant2l 769 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
2015, 17, 193eqtr3d 2547 . . . . . . . . . 10 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
2120exp31 621 . . . . . . . . 9 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (𝑥𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
2213, 21ralrimi 2835 . . . . . . . 8 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23 ralim 2827 . . . . . . . 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 228 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2625expd 445 . . . . 5 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → (∀𝑥𝐴 𝐶 ∈ V → ∀𝑥𝐴 𝐵 = 𝐶)))
279, 26mpdd 41 . . . 4 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → ∀𝑥𝐴 𝐵 = 𝐶))
2827com12 32 . . 3 (∀𝑥𝐴 𝐵 ∈ V → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 𝐵 = 𝐶))
29 eqid 2505 . . . 4 𝐴 = 𝐴
30 mpteq12 4514 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3129, 30mpan 693 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3228, 31impbid1 210 . 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 191  wa 378   = wceq 1468  wcel 1937  wral 2791  Vcvv 3066  cmpt 4493   Fn wfn 5628  cfv 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3758  df-if 3909  df-sn 3996  df-pr 3998  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-mpt 4495  df-id 4795  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5597  df-fun 5635  df-fn 5636  df-fv 5641
This theorem is referenced by:  eqfnfv  6043  eufnfv  6209  offveqb  6629  ramcl  15149  fucsect  16043  setcepi  16149  0frgp  17590  dprdf11  17816  dpjeq  17852  mvrf1  18808  mplmonmul  18847  frgpcyg  19302  ustuqtop  21419  mdegle0  23187  ply1nzb  23232  cvmliftphtlem  30192  matunitlindflem1  32174
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