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Theorem fmptcos 5275
 Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fmptcof.1 (φx A 𝑅 B)
fmptcof.2 (φ𝐹 = (x A𝑅))
fmptcof.3 (φ𝐺 = (y B𝑆))
Assertion
Ref Expression
fmptcos (φ → (𝐺𝐹) = (x A𝑅 / y𝑆))
Distinct variable groups:   x,y,B   y,𝑅   x,𝑆   x,A
Allowed substitution hints:   φ(x,y)   A(y)   𝑅(x)   𝑆(y)   𝐹(x,y)   𝐺(x,y)

Proof of Theorem fmptcos
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 fmptcof.1 . 2 (φx A 𝑅 B)
2 fmptcof.2 . 2 (φ𝐹 = (x A𝑅))
3 fmptcof.3 . . 3 (φ𝐺 = (y B𝑆))
4 nfcv 2175 . . . 4 z𝑆
5 nfcsb1v 2876 . . . 4 yz / y𝑆
6 csbeq1a 2854 . . . 4 (y = z𝑆 = z / y𝑆)
74, 5, 6cbvmpt 3842 . . 3 (y B𝑆) = (z Bz / y𝑆)
83, 7syl6eq 2085 . 2 (φ𝐺 = (z Bz / y𝑆))
9 csbeq1 2849 . 2 (z = 𝑅z / y𝑆 = 𝑅 / y𝑆)
101, 2, 8, 9fmptcof 5274 1 (φ → (𝐺𝐹) = (x A𝑅 / y𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ⦋csb 2846   ↦ cmpt 3809   ∘ ccom 4292 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-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 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-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 This theorem is referenced by:  fmpt2co  5779
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