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Theorem brcofffn 40259
Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)
Hypotheses
Ref Expression
brcofffn.c (𝜑𝐶 Fn 𝑍)
brcofffn.d (𝜑𝐷:𝑌𝑍)
brcofffn.e (𝜑𝐸:𝑋𝑌)
brcofffn.r (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
Assertion
Ref Expression
brcofffn (𝜑 → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))

Proof of Theorem brcofffn
StepHypRef Expression
1 brcofffn.c . . . . 5 (𝜑𝐶 Fn 𝑍)
2 brcofffn.d . . . . 5 (𝜑𝐷:𝑌𝑍)
3 fnfco 6536 . . . . 5 ((𝐶 Fn 𝑍𝐷:𝑌𝑍) → (𝐶𝐷) Fn 𝑌)
41, 2, 3syl2anc 584 . . . 4 (𝜑 → (𝐶𝐷) Fn 𝑌)
5 brcofffn.e . . . 4 (𝜑𝐸:𝑋𝑌)
6 brcofffn.r . . . . 5 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
7 coass 6111 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
87breqi 5063 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
96, 8sylibr 235 . . . 4 (𝜑𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
104, 5, 9brcoffn 40258 . . 3 (𝜑 → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵))
111adantr 481 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → 𝐶 Fn 𝑍)
122adantr 481 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → 𝐷:𝑌𝑍)
13 simprr 769 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → (𝐸𝐴)(𝐶𝐷)𝐵)
1411, 12, 13brcoffn 40258 . . . 4 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))
1514ex 413 . . 3 (𝜑 → ((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) → ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)))
1610, 15jcai 517 . 2 (𝜑 → ((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)))
17 simpll 763 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → 𝐴𝐸(𝐸𝐴))
18 simprl 767 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)))
19 simprr 769 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐷‘(𝐸𝐴))𝐶𝐵)
2017, 18, 193jca 1120 . 2 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))
2116, 20syl 17 1 (𝜑 → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   class class class wbr 5057  ccom 5552   Fn wfn 6343  wf 6344  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  brco3f1o  40261  neicvgmex  40345  neicvgel1  40347
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