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Theorem oprab2co 5781
Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
Hypotheses
Ref Expression
oprab2co.1 ((x A y B) → 𝐶 𝑅)
oprab2co.2 ((x A y B) → 𝐷 𝑆)
oprab2co.3 𝐹 = (x A, y B ↦ ⟨𝐶, 𝐷⟩)
oprab2co.4 𝐺 = (x A, y B ↦ (𝐶𝑀𝐷))
Assertion
Ref Expression
oprab2co (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Distinct variable groups:   x,y,A   x,B,y   x,𝑀,y   x,𝑅,y   x,𝑆,y
Allowed substitution hints:   𝐶(x,y)   𝐷(x,y)   𝐹(x,y)   𝐺(x,y)

Proof of Theorem oprab2co
StepHypRef Expression
1 oprab2co.1 . . 3 ((x A y B) → 𝐶 𝑅)
2 oprab2co.2 . . 3 ((x A y B) → 𝐷 𝑆)
3 opelxpi 4319 . . 3 ((𝐶 𝑅 𝐷 𝑆) → ⟨𝐶, 𝐷 (𝑅 × 𝑆))
41, 2, 3syl2anc 391 . 2 ((x A y B) → ⟨𝐶, 𝐷 (𝑅 × 𝑆))
5 oprab2co.3 . 2 𝐹 = (x A, y B ↦ ⟨𝐶, 𝐷⟩)
6 oprab2co.4 . . 3 𝐺 = (x A, y B ↦ (𝐶𝑀𝐷))
7 df-ov 5458 . . . . 5 (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
87a1i 9 . . . 4 ((x A y B) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
98mpt2eq3ia 5512 . . 3 (x A, y B ↦ (𝐶𝑀𝐷)) = (x A, y B ↦ (𝑀‘⟨𝐶, 𝐷⟩))
106, 9eqtri 2057 . 2 𝐺 = (x A, y B ↦ (𝑀‘⟨𝐶, 𝐷⟩))
114, 5, 10oprabco 5780 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cop 3370   × cxp 4286  ccom 4292   Fn wfn 4840  cfv 4845  (class class class)co 5455  cmpt2 5457
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-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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