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Theorem mpteqb 5607
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5614. (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 2749 . . 3 (𝐵𝑉𝐵 ∈ V)
21ralimi 2540 . 2 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵 ∈ V)
3 fneq1 5305 . . . . . . 7 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((𝑥𝐴𝐵) Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
4 eqid 2177 . . . . . . . 8 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54mptfng 5342 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
6 eqid 2177 . . . . . . . 8 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
76mptfng 5342 . . . . . . 7 (∀𝑥𝐴 𝐶 ∈ V ↔ (𝑥𝐴𝐶) Fn 𝐴)
83, 5, 73bitr4g 223 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V ↔ ∀𝑥𝐴 𝐶 ∈ V))
98biimpd 144 . . . . 5 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 𝐵 ∈ V → ∀𝑥𝐴 𝐶 ∈ V))
10 r19.26 2603 . . . . . . 7 (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V))
11 nfmpt1 4097 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
12 nfmpt1 4097 . . . . . . . . . 10 𝑥(𝑥𝐴𝐶)
1311, 12nfeq 2327 . . . . . . . . 9 𝑥(𝑥𝐴𝐵) = (𝑥𝐴𝐶)
14 simpll 527 . . . . . . . . . . . 12 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
1514fveq1d 5518 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = ((𝑥𝐴𝐶)‘𝑥))
164fvmpt2 5600 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ V) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1716ad2ant2lr 510 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
186fvmpt2 5600 . . . . . . . . . . . 12 ((𝑥𝐴𝐶 ∈ V) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
1918ad2ant2l 508 . . . . . . . . . . 11 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
2015, 17, 193eqtr3d 2218 . . . . . . . . . 10 ((((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ∧ 𝑥𝐴) ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 = 𝐶)
2120exp31 364 . . . . . . . . 9 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (𝑥𝐴 → ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶)))
2213, 21ralrimi 2548 . . . . . . . 8 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶))
23 ralim 2536 . . . . . . . 8 (∀𝑥𝐴 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐵 = 𝐶) → (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2422, 23syl 14 . . . . . . 7 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → (∀𝑥𝐴 (𝐵 ∈ V ∧ 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2510, 24biimtrrid 153 . . . . . 6 ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ((∀𝑥𝐴 𝐵 ∈ V ∧ ∀𝑥𝐴 𝐶 ∈ V) → ∀𝑥𝐴 𝐵 = 𝐶))
2625expd 258 . . . . 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 30 . . 3 (∀𝑥𝐴 𝐵 ∈ V → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) → ∀𝑥𝐴 𝐵 = 𝐶))
29 eqid 2177 . . . 4 𝐴 = 𝐴
30 mpteq12 4087 . . . 4 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3129, 30mpan 424 . . 3 (∀𝑥𝐴 𝐵 = 𝐶 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
3228, 31impbid1 142 . 2 (∀𝑥𝐴 𝐵 ∈ V → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
332, 32syl 14 1 (∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  Vcvv 2738  cmpt 4065   Fn wfn 5212  cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225
This theorem is referenced by:  eqfnfv  5614  eufnfv  5748  offveqb  6102
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