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Theorem fmpt2x 5745
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 2534 . . . . . . . 8 z V
2 vex 2534 . . . . . . . 8 w V
31, 2op1std 5694 . . . . . . 7 (v = ⟨z, w⟩ → (1stv) = z)
43csbeq1d 2831 . . . . . 6 (v = ⟨z, w⟩ → (1stv) / x(2ndv) / y𝐶 = z / x(2ndv) / y𝐶)
51, 2op2ndd 5695 . . . . . . . 8 (v = ⟨z, w⟩ → (2ndv) = w)
65csbeq1d 2831 . . . . . . 7 (v = ⟨z, w⟩ → (2ndv) / y𝐶 = w / y𝐶)
76csbeq2dv 2848 . . . . . 6 (v = ⟨z, w⟩ → z / x(2ndv) / y𝐶 = z / xw / y𝐶)
84, 7eqtrd 2050 . . . . 5 (v = ⟨z, w⟩ → (1stv) / x(2ndv) / y𝐶 = z / xw / y𝐶)
98eleq1d 2084 . . . 4 (v = ⟨z, w⟩ → ((1stv) / x(2ndv) / y𝐶 𝐷z / xw / y𝐶 𝐷))
109raliunxp 4400 . . 3 (v z A ({z} × z / xB)(1stv) / x(2ndv) / y𝐶 𝐷z A w z / xBz / xw / y𝐶 𝐷)
11 nfv 1398 . . . . . . 7 z((x A y B) v = 𝐶)
12 nfv 1398 . . . . . . 7 w((x A y B) v = 𝐶)
13 nfv 1398 . . . . . . . . 9 x z A
14 nfcsb1v 2855 . . . . . . . . . 10 xz / xB
1514nfcri 2150 . . . . . . . . 9 x w z / xB
1613, 15nfan 1435 . . . . . . . 8 x(z A w z / xB)
17 nfcsb1v 2855 . . . . . . . . 9 xz / xw / y𝐶
1817nfeq2 2167 . . . . . . . 8 x v = z / xw / y𝐶
1916, 18nfan 1435 . . . . . . 7 x((z A w z / xB) v = z / xw / y𝐶)
20 nfv 1398 . . . . . . . 8 y(z A w z / xB)
21 nfcv 2156 . . . . . . . . . 10 yz
22 nfcsb1v 2855 . . . . . . . . . 10 yw / y𝐶
2321, 22nfcsb 2857 . . . . . . . . 9 yz / xw / y𝐶
2423nfeq2 2167 . . . . . . . 8 y v = z / xw / y𝐶
2520, 24nfan 1435 . . . . . . 7 y((z A w z / xB) v = z / xw / y𝐶)
26 eleq1 2078 . . . . . . . . . 10 (x = z → (x Az A))
2726adantr 261 . . . . . . . . 9 ((x = z y = w) → (x Az A))
28 eleq1 2078 . . . . . . . . . 10 (y = w → (y Bw B))
29 csbeq1a 2833 . . . . . . . . . . 11 (x = zB = z / xB)
3029eleq2d 2085 . . . . . . . . . 10 (x = z → (w Bw z / xB))
3128, 30sylan9bbr 439 . . . . . . . . 9 ((x = z y = w) → (y Bw z / xB))
3227, 31anbi12d 445 . . . . . . . 8 ((x = z y = w) → ((x A y B) ↔ (z A w z / xB)))
33 csbeq1a 2833 . . . . . . . . . 10 (y = w𝐶 = w / y𝐶)
34 csbeq1a 2833 . . . . . . . . . 10 (x = zw / y𝐶 = z / xw / y𝐶)
3533, 34sylan9eqr 2072 . . . . . . . . 9 ((x = z y = w) → 𝐶 = z / xw / y𝐶)
3635eqeq2d 2029 . . . . . . . 8 ((x = z y = w) → (v = 𝐶v = z / xw / y𝐶))
3732, 36anbi12d 445 . . . . . . 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 5497 . . . . . 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 5437 . . . . . 6 (x A, y B𝐶) = {⟨⟨x, y⟩, v⟩ ∣ ((x A y B) v = 𝐶)}
40 df-mpt2 5437 . . . . . 6 (z A, w z / xBz / xw / y𝐶) = {⟨⟨z, w⟩, v⟩ ∣ ((z A w z / xB) v = z / xw / y𝐶)}
4138, 39, 403eqtr4i 2048 . . . . 5 (x A, y B𝐶) = (z A, w z / xBz / xw / y𝐶)
42 fmpt2x.1 . . . . 5 𝐹 = (x A, y B𝐶)
438mpt2mptx 5514 . . . . 5 (v z A ({z} × z / xB) ↦ (1stv) / x(2ndv) / y𝐶) = (z A, w z / xBz / xw / y𝐶)
4441, 42, 433eqtr4i 2048 . . . 4 𝐹 = (v z A ({z} × z / xB) ↦ (1stv) / x(2ndv) / y𝐶)
4544fmpt 5240 . . 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 1398 . . 3 zy B 𝐶 𝐷
4817nfel1 2166 . . . 4 xz / xw / y𝐶 𝐷
4914, 48nfralxy 2334 . . 3 xw z / xBz / xw / y𝐶 𝐷
50 nfv 1398 . . . . 5 w 𝐶 𝐷
5122nfel1 2166 . . . . 5 yw / y𝐶 𝐷
5233eleq1d 2084 . . . . 5 (y = w → (𝐶 𝐷w / y𝐶 𝐷))
5350, 51, 52cbvral 2503 . . . 4 (y B 𝐶 𝐷w B w / y𝐶 𝐷)
5434eleq1d 2084 . . . . 5 (x = z → (w / y𝐶 𝐷z / xw / y𝐶 𝐷))
5529, 54raleqbidv 2491 . . . 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 2503 . 2 (x A y B 𝐶 𝐷z A w z / xBz / xw / y𝐶 𝐷)
58 nfcv 2156 . . . 4 z({x} × B)
59 nfcv 2156 . . . . 5 x{z}
6059, 14nfxp 4294 . . . 4 x({z} × z / xB)
61 sneq 3357 . . . . 5 (x = z → {x} = {z})
6261, 29xpeq12d 4293 . . . 4 (x = z → ({x} × B) = ({z} × z / xB))
6358, 60, 62cbviun 3664 . . 3 x A ({x} × B) = z A ({z} × z / xB)
6463feq2i 4962 . 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 1226   wcel 1370  wral 2280  csb 2825  {csn 3346  cop 3349   ciun 3627  cmpt 3788   × cxp 4266  wf 4821  cfv 4825  {coprab 5433  cmpt2 5434  1st c1st 5684  2nd c2nd 5685
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-fv 4833  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687
This theorem is referenced by:  fmpt2  5746
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