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Theorem fmpt2x 5768
Description: Functionality, domain and codomain of a class given by the "maps to" notation, where B(x) is not constant but depends on x. (Contributed by NM, 29-Dec-2014.)
Hypothesis
Ref Expression
fmpt2x.1 𝐹 = (x A, y B𝐶)
Assertion
Ref Expression
fmpt2x (x A y B 𝐶 𝐷𝐹: x A ({x} × B)⟶𝐷)
Distinct variable groups:   x,y,A   y,B   x,𝐷,y
Allowed substitution hints:   B(x)   𝐶(x,y)   𝐹(x,y)

Proof of Theorem fmpt2x
Dummy variables v w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . 8 z V
2 vex 2554 . . . . . . . 8 w V
31, 2op1std 5717 . . . . . . 7 (v = ⟨z, w⟩ → (1stv) = z)
43csbeq1d 2852 . . . . . 6 (v = ⟨z, w⟩ → (1stv) / x(2ndv) / y𝐶 = z / x(2ndv) / y𝐶)
51, 2op2ndd 5718 . . . . . . . 8 (v = ⟨z, w⟩ → (2ndv) = w)
65csbeq1d 2852 . . . . . . 7 (v = ⟨z, w⟩ → (2ndv) / y𝐶 = w / y𝐶)
76csbeq2dv 2869 . . . . . 6 (v = ⟨z, w⟩ → z / x(2ndv) / y𝐶 = z / xw / y𝐶)
84, 7eqtrd 2069 . . . . 5 (v = ⟨z, w⟩ → (1stv) / x(2ndv) / y𝐶 = z / xw / y𝐶)
98eleq1d 2103 . . . 4 (v = ⟨z, w⟩ → ((1stv) / x(2ndv) / y𝐶 𝐷z / xw / y𝐶 𝐷))
109raliunxp 4420 . . 3 (v z A ({z} × z / xB)(1stv) / x(2ndv) / y𝐶 𝐷z A w z / xBz / xw / y𝐶 𝐷)
11 nfv 1418 . . . . . . 7 z((x A y B) v = 𝐶)
12 nfv 1418 . . . . . . 7 w((x A y B) v = 𝐶)
13 nfv 1418 . . . . . . . . 9 x z A
14 nfcsb1v 2876 . . . . . . . . . 10 xz / xB
1514nfcri 2169 . . . . . . . . 9 x w z / xB
1613, 15nfan 1454 . . . . . . . 8 x(z A w z / xB)
17 nfcsb1v 2876 . . . . . . . . 9 xz / xw / y𝐶
1817nfeq2 2186 . . . . . . . 8 x v = z / xw / y𝐶
1916, 18nfan 1454 . . . . . . 7 x((z A w z / xB) v = z / xw / y𝐶)
20 nfv 1418 . . . . . . . 8 y(z A w z / xB)
21 nfcv 2175 . . . . . . . . . 10 yz
22 nfcsb1v 2876 . . . . . . . . . 10 yw / y𝐶
2321, 22nfcsb 2878 . . . . . . . . 9 yz / xw / y𝐶
2423nfeq2 2186 . . . . . . . 8 y v = z / xw / y𝐶
2520, 24nfan 1454 . . . . . . 7 y((z A w z / xB) v = z / xw / y𝐶)
26 eleq1 2097 . . . . . . . . . 10 (x = z → (x Az A))
2726adantr 261 . . . . . . . . 9 ((x = z y = w) → (x Az A))
28 eleq1 2097 . . . . . . . . . 10 (y = w → (y Bw B))
29 csbeq1a 2854 . . . . . . . . . . 11 (x = zB = z / xB)
3029eleq2d 2104 . . . . . . . . . 10 (x = z → (w Bw z / xB))
3128, 30sylan9bbr 436 . . . . . . . . 9 ((x = z y = w) → (y Bw z / xB))
3227, 31anbi12d 442 . . . . . . . 8 ((x = z y = w) → ((x A y B) ↔ (z A w z / xB)))
33 csbeq1a 2854 . . . . . . . . . 10 (y = w𝐶 = w / y𝐶)
34 csbeq1a 2854 . . . . . . . . . 10 (x = zw / y𝐶 = z / xw / y𝐶)
3533, 34sylan9eqr 2091 . . . . . . . . 9 ((x = z y = w) → 𝐶 = z / xw / y𝐶)
3635eqeq2d 2048 . . . . . . . 8 ((x = z y = w) → (v = 𝐶v = z / xw / y𝐶))
3732, 36anbi12d 442 . . . . . . 7 ((x = z y = w) → (((x A y B) v = 𝐶) ↔ ((z A w z / xB) v = z / xw / y𝐶)))
3811, 12, 19, 25, 37cbvoprab12 5520 . . . . . 6 {⟨⟨x, y⟩, v⟩ ∣ ((x A y B) v = 𝐶)} = {⟨⟨z, w⟩, v⟩ ∣ ((z A w z / xB) v = z / xw / y𝐶)}
39 df-mpt2 5460 . . . . . 6 (x A, y B𝐶) = {⟨⟨x, y⟩, v⟩ ∣ ((x A y B) v = 𝐶)}
40 df-mpt2 5460 . . . . . 6 (z A, w z / xBz / xw / y𝐶) = {⟨⟨z, w⟩, v⟩ ∣ ((z A w z / xB) v = z / xw / y𝐶)}
4138, 39, 403eqtr4i 2067 . . . . 5 (x A, y B𝐶) = (z A, w z / xBz / xw / y𝐶)
42 fmpt2x.1 . . . . 5 𝐹 = (x A, y B𝐶)
438mpt2mptx 5537 . . . . 5 (v z A ({z} × z / xB) ↦ (1stv) / x(2ndv) / y𝐶) = (z A, w z / xBz / xw / y𝐶)
4441, 42, 433eqtr4i 2067 . . . 4 𝐹 = (v z A ({z} × z / xB) ↦ (1stv) / x(2ndv) / y𝐶)
4544fmpt 5262 . . 3 (v z A ({z} × z / xB)(1stv) / x(2ndv) / y𝐶 𝐷𝐹: z A ({z} × z / xB)⟶𝐷)
4610, 45bitr3i 175 . 2 (z A w z / xBz / xw / y𝐶 𝐷𝐹: z A ({z} × z / xB)⟶𝐷)
47 nfv 1418 . . 3 zy B 𝐶 𝐷
4817nfel1 2185 . . . 4 xz / xw / y𝐶 𝐷
4914, 48nfralxy 2354 . . 3 xw z / xBz / xw / y𝐶 𝐷
50 nfv 1418 . . . . 5 w 𝐶 𝐷
5122nfel1 2185 . . . . 5 yw / y𝐶 𝐷
5233eleq1d 2103 . . . . 5 (y = w → (𝐶 𝐷w / y𝐶 𝐷))
5350, 51, 52cbvral 2523 . . . 4 (y B 𝐶 𝐷w B w / y𝐶 𝐷)
5434eleq1d 2103 . . . . 5 (x = z → (w / y𝐶 𝐷z / xw / y𝐶 𝐷))
5529, 54raleqbidv 2511 . . . 4 (x = z → (w B w / y𝐶 𝐷w z / xBz / xw / y𝐶 𝐷))
5653, 55syl5bb 181 . . 3 (x = z → (y B 𝐶 𝐷w z / xBz / xw / y𝐶 𝐷))
5747, 49, 56cbvral 2523 . 2 (x A y B 𝐶 𝐷z A w z / xBz / xw / y𝐶 𝐷)
58 nfcv 2175 . . . 4 z({x} × B)
59 nfcv 2175 . . . . 5 x{z}
6059, 14nfxp 4314 . . . 4 x({z} × z / xB)
61 sneq 3378 . . . . 5 (x = z → {x} = {z})
6261, 29xpeq12d 4313 . . . 4 (x = z → ({x} × B) = ({z} × z / xB))
6358, 60, 62cbviun 3685 . . 3 x A ({x} × B) = z A ({z} × z / xB)
6463feq2i 4983 . 2 (𝐹: x A ({x} × B)⟶𝐷𝐹: z A ({z} × z / xB)⟶𝐷)
6546, 57, 643bitr4i 201 1 (x A y B 𝐶 𝐷𝐹: x A ({x} × B)⟶𝐷)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  csb 2846  {csn 3367  cop 3370   ciun 3648  cmpt 3809   × cxp 4286  wf 4841  cfv 4845  {coprab 5456  cmpt2 5457  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  fmpt2  5769
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