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Theorem brcofffn 41318
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 6584 . . . . 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 6129 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
87breqi 5059 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
96, 8sylibr 237 . . . 4 (𝜑𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
104, 5, 9brcoffn 41317 . . 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 773 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → (𝐸𝐴)(𝐶𝐷)𝐵)
1411, 12, 13brcoffn 41317 . . . 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 767 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → 𝐴𝐸(𝐸𝐴))
18 simprl 771 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)))
19 simprr 773 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐷‘(𝐸𝐴))𝐶𝐵)
2017, 18, 193jca 1130 . 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 1089   class class class wbr 5053  ccom 5555   Fn wfn 6375  wf 6376  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388
This theorem is referenced by:  brco3f1o  41320  neicvgmex  41404  neicvgel1  41406
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