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Theorem brcofffn 42245
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 6704 . . . . 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 6215 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
87breqi 5109 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
96, 8sylibr 233 . . . 4 (𝜑𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
104, 5, 9brcoffn 42244 . . 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 771 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → (𝐸𝐴)(𝐶𝐷)𝐵)
1411, 12, 13brcoffn 42244 . . . 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 765 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → 𝐴𝐸(𝐸𝐴))
18 simprl 769 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)))
19 simprr 771 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐷‘(𝐸𝐴))𝐶𝐵)
2017, 18, 193jca 1128 . 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 1087   class class class wbr 5103  ccom 5635   Fn wfn 6488  wf 6489  cfv 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501
This theorem is referenced by:  brco3f1o  42247  neicvgmex  42331  neicvgel1  42333
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