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Theorem fmpt2co 5779
Description: Composition of two functions. Variation of fmptco 5273 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
fmpt2co.1 ((φ (x A y B)) → 𝑅 𝐶)
fmpt2co.2 (φ𝐹 = (x A, y B𝑅))
fmpt2co.3 (φ𝐺 = (z 𝐶𝑆))
fmpt2co.4 (z = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
fmpt2co (φ → (𝐺𝐹) = (x A, y B𝑇))
Distinct variable groups:   x,y,B   x,z,𝐶,y   φ,x,y   x,𝑆,y   x,A,y   z,𝑅   z,𝑇
Allowed substitution hints:   φ(z)   A(z)   B(z)   𝑅(x,y)   𝑆(z)   𝑇(x,y)   𝐹(x,y,z)   𝐺(x,y,z)

Proof of Theorem fmpt2co
Dummy variables u v w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmpt2co.1 . . . . . 6 ((φ (x A y B)) → 𝑅 𝐶)
21ralrimivva 2395 . . . . 5 (φx A y B 𝑅 𝐶)
3 eqid 2037 . . . . . 6 (x A, y B𝑅) = (x A, y B𝑅)
43fmpt2 5769 . . . . 5 (x A y B 𝑅 𝐶 ↔ (x A, y B𝑅):(A × B)⟶𝐶)
52, 4sylib 127 . . . 4 (φ → (x A, y B𝑅):(A × B)⟶𝐶)
6 nfcv 2175 . . . . . . 7 u𝑅
7 nfcv 2175 . . . . . . 7 v𝑅
8 nfcv 2175 . . . . . . . 8 xv
9 nfcsb1v 2876 . . . . . . . 8 xu / x𝑅
108, 9nfcsb 2878 . . . . . . 7 xv / yu / x𝑅
11 nfcsb1v 2876 . . . . . . 7 yv / yu / x𝑅
12 csbeq1a 2854 . . . . . . . 8 (x = u𝑅 = u / x𝑅)
13 csbeq1a 2854 . . . . . . . 8 (y = vu / x𝑅 = v / yu / x𝑅)
1412, 13sylan9eq 2089 . . . . . . 7 ((x = u y = v) → 𝑅 = v / yu / x𝑅)
156, 7, 10, 11, 14cbvmpt2 5525 . . . . . 6 (x A, y B𝑅) = (u A, v Bv / yu / x𝑅)
16 vex 2554 . . . . . . . . . 10 u V
17 vex 2554 . . . . . . . . . 10 v V
1816, 17op2ndd 5718 . . . . . . . . 9 (w = ⟨u, v⟩ → (2ndw) = v)
1918csbeq1d 2852 . . . . . . . 8 (w = ⟨u, v⟩ → (2ndw) / y(1stw) / x𝑅 = v / y(1stw) / x𝑅)
2016, 17op1std 5717 . . . . . . . . . 10 (w = ⟨u, v⟩ → (1stw) = u)
2120csbeq1d 2852 . . . . . . . . 9 (w = ⟨u, v⟩ → (1stw) / x𝑅 = u / x𝑅)
2221csbeq2dv 2869 . . . . . . . 8 (w = ⟨u, v⟩ → v / y(1stw) / x𝑅 = v / yu / x𝑅)
2319, 22eqtrd 2069 . . . . . . 7 (w = ⟨u, v⟩ → (2ndw) / y(1stw) / x𝑅 = v / yu / x𝑅)
2423mpt2mpt 5538 . . . . . 6 (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅) = (u A, v Bv / yu / x𝑅)
2515, 24eqtr4i 2060 . . . . 5 (x A, y B𝑅) = (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅)
2625fmpt 5262 . . . 4 (w (A × B)(2ndw) / y(1stw) / x𝑅 𝐶 ↔ (x A, y B𝑅):(A × B)⟶𝐶)
275, 26sylibr 137 . . 3 (φw (A × B)(2ndw) / y(1stw) / x𝑅 𝐶)
28 fmpt2co.2 . . . 4 (φ𝐹 = (x A, y B𝑅))
2928, 25syl6eq 2085 . . 3 (φ𝐹 = (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅))
30 fmpt2co.3 . . 3 (φ𝐺 = (z 𝐶𝑆))
3127, 29, 30fmptcos 5275 . 2 (φ → (𝐺𝐹) = (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅 / z𝑆))
3223csbeq1d 2852 . . . . 5 (w = ⟨u, v⟩ → (2ndw) / y(1stw) / x𝑅 / z𝑆 = v / yu / x𝑅 / z𝑆)
3332mpt2mpt 5538 . . . 4 (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅 / z𝑆) = (u A, v Bv / yu / x𝑅 / z𝑆)
34 nfcv 2175 . . . . 5 u𝑅 / z𝑆
35 nfcv 2175 . . . . 5 v𝑅 / z𝑆
36 nfcv 2175 . . . . . 6 x𝑆
3710, 36nfcsb 2878 . . . . 5 xv / yu / x𝑅 / z𝑆
38 nfcv 2175 . . . . . 6 y𝑆
3911, 38nfcsb 2878 . . . . 5 yv / yu / x𝑅 / z𝑆
4014csbeq1d 2852 . . . . 5 ((x = u y = v) → 𝑅 / z𝑆 = v / yu / x𝑅 / z𝑆)
4134, 35, 37, 39, 40cbvmpt2 5525 . . . 4 (x A, y B𝑅 / z𝑆) = (u A, v Bv / yu / x𝑅 / z𝑆)
4233, 41eqtr4i 2060 . . 3 (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅 / z𝑆) = (x A, y B𝑅 / z𝑆)
4313impb 1099 . . . . 5 ((φ x A y B) → 𝑅 𝐶)
44 nfcvd 2176 . . . . . 6 (𝑅 𝐶z𝑇)
45 fmpt2co.4 . . . . . 6 (z = 𝑅𝑆 = 𝑇)
4644, 45csbiegf 2884 . . . . 5 (𝑅 𝐶𝑅 / z𝑆 = 𝑇)
4743, 46syl 14 . . . 4 ((φ x A y B) → 𝑅 / z𝑆 = 𝑇)
4847mpt2eq3dva 5511 . . 3 (φ → (x A, y B𝑅 / z𝑆) = (x A, y B𝑇))
4942, 48syl5eq 2081 . 2 (φ → (w (A × B) ↦ (2ndw) / y(1stw) / x𝑅 / z𝑆) = (x A, y B𝑇))
5031, 49eqtrd 2069 1 (φ → (𝐺𝐹) = (x A, y B𝑇))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wral 2300  csb 2846  cop 3370  cmpt 3809   × cxp 4286  ccom 4292  wf 4841  cfv 4845  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:  oprabco  5780
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