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Theorem mpteqb 5186
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5190. (Contributed by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
mpteqb (x A B 𝑉 → ((x AB) = (x A𝐶) ↔ x A B = 𝐶))
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝐶(x)   𝑉(x)

Proof of Theorem mpteqb
StepHypRef Expression
1 elex 2543 . . 3 (B 𝑉B V)
21ralimi 2362 . 2 (x A B 𝑉x A B V)
3 fneq1 4913 . . . . . . 7 ((x AB) = (x A𝐶) → ((x AB) Fn A ↔ (x A𝐶) Fn A))
4 eqid 2022 . . . . . . . 8 (x AB) = (x AB)
54mptfng 4950 . . . . . . 7 (x A B V ↔ (x AB) Fn A)
6 eqid 2022 . . . . . . . 8 (x A𝐶) = (x A𝐶)
76mptfng 4950 . . . . . . 7 (x A 𝐶 V ↔ (x A𝐶) Fn A)
83, 5, 73bitr4g 212 . . . . . 6 ((x AB) = (x A𝐶) → (x A B V ↔ x A 𝐶 V))
98biimpd 132 . . . . 5 ((x AB) = (x A𝐶) → (x A B V → x A 𝐶 V))
10 r19.26 2419 . . . . . . 7 (x A (B V 𝐶 V) ↔ (x A B V x A 𝐶 V))
11 nfmpt1 3824 . . . . . . . . . 10 x(x AB)
12 nfmpt1 3824 . . . . . . . . . 10 x(x A𝐶)
1311, 12nfeq 2167 . . . . . . . . 9 x(x AB) = (x A𝐶)
14 simpll 469 . . . . . . . . . . . 12 ((((x AB) = (x A𝐶) x A) (B V 𝐶 V)) → (x AB) = (x A𝐶))
1514fveq1d 5105 . . . . . . . . . . 11 ((((x AB) = (x A𝐶) x A) (B V 𝐶 V)) → ((x AB)‘x) = ((x A𝐶)‘x))
164fvmpt2 5179 . . . . . . . . . . . 12 ((x A B V) → ((x AB)‘x) = B)
1716ad2ant2lr 467 . . . . . . . . . . 11 ((((x AB) = (x A𝐶) x A) (B V 𝐶 V)) → ((x AB)‘x) = B)
186fvmpt2 5179 . . . . . . . . . . . 12 ((x A 𝐶 V) → ((x A𝐶)‘x) = 𝐶)
1918ad2ant2l 465 . . . . . . . . . . 11 ((((x AB) = (x A𝐶) x A) (B V 𝐶 V)) → ((x A𝐶)‘x) = 𝐶)
2015, 17, 193eqtr3d 2062 . . . . . . . . . 10 ((((x AB) = (x A𝐶) x A) (B V 𝐶 V)) → B = 𝐶)
2120exp31 346 . . . . . . . . 9 ((x AB) = (x A𝐶) → (x A → ((B V 𝐶 V) → B = 𝐶)))
2213, 21ralrimi 2368 . . . . . . . 8 ((x AB) = (x A𝐶) → x A ((B V 𝐶 V) → B = 𝐶))
23 ralim 2358 . . . . . . . 8 (x A ((B V 𝐶 V) → B = 𝐶) → (x A (B V 𝐶 V) → x A B = 𝐶))
2422, 23syl 14 . . . . . . 7 ((x AB) = (x A𝐶) → (x A (B V 𝐶 V) → x A B = 𝐶))
2510, 24syl5bir 142 . . . . . 6 ((x AB) = (x A𝐶) → ((x A B V x A 𝐶 V) → x A B = 𝐶))
2625expd 245 . . . . 5 ((x AB) = (x A𝐶) → (x A B V → (x A 𝐶 V → x A B = 𝐶)))
279, 26mpdd 36 . . . 4 ((x AB) = (x A𝐶) → (x A B V → x A B = 𝐶))
2827com12 27 . . 3 (x A B V → ((x AB) = (x A𝐶) → x A B = 𝐶))
29 eqid 2022 . . . 4 A = A
30 mpteq12 3814 . . . 4 ((A = A x A B = 𝐶) → (x AB) = (x A𝐶))
3129, 30mpan 402 . . 3 (x A B = 𝐶 → (x AB) = (x A𝐶))
3228, 31impbid1 130 . 2 (x A B V → ((x AB) = (x A𝐶) ↔ x A B = 𝐶))
332, 32syl 14 1 (x A B 𝑉 → ((x AB) = (x A𝐶) ↔ x A B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wral 2284  Vcvv 2535  cmpt 3792   Fn wfn 4824  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  eqfnfv  5190  eufnfv  5314  offveqb  5653
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