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Theorem brcofffn 40777
 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 6518 . . . . 5 ((𝐶 Fn 𝑍𝐷:𝑌𝑍) → (𝐶𝐷) Fn 𝑌)
41, 2, 3syl2anc 587 . . . 4 (𝜑 → (𝐶𝐷) Fn 𝑌)
5 brcofffn.e . . . 4 (𝜑𝐸:𝑋𝑌)
6 brcofffn.r . . . . 5 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
7 coass 6086 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
87breqi 5037 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
96, 8sylibr 237 . . . 4 (𝜑𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
104, 5, 9brcoffn 40776 . . 3 (𝜑 → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵))
111adantr 484 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → 𝐶 Fn 𝑍)
122adantr 484 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → 𝐷:𝑌𝑍)
13 simprr 772 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → (𝐸𝐴)(𝐶𝐷)𝐵)
1411, 12, 13brcoffn 40776 . . . 4 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))
1514ex 416 . . 3 (𝜑 → ((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) → ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)))
1610, 15jcai 520 . 2 (𝜑 → ((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)))
17 simpll 766 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → 𝐴𝐸(𝐸𝐴))
18 simprl 770 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)))
19 simprr 772 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐷‘(𝐸𝐴))𝐶𝐵)
2017, 18, 193jca 1125 . 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 399   ∧ w3a 1084   class class class wbr 5031   ∘ ccom 5524   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333 This theorem is referenced by:  brco3f1o  40779  neicvgmex  40863  neicvgel1  40865
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