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Theorem brcofffn 38831
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 6230 . . . . 5 ((𝐶 Fn 𝑍𝐷:𝑌𝑍) → (𝐶𝐷) Fn 𝑌)
41, 2, 3syl2anc 696 . . . 4 (𝜑 → (𝐶𝐷) Fn 𝑌)
5 brcofffn.e . . . 4 (𝜑𝐸:𝑋𝑌)
6 brcofffn.r . . . . 5 (𝜑𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
7 coass 5815 . . . . . 6 ((𝐶𝐷) ∘ 𝐸) = (𝐶 ∘ (𝐷𝐸))
87breqi 4810 . . . . 5 (𝐴((𝐶𝐷) ∘ 𝐸)𝐵𝐴(𝐶 ∘ (𝐷𝐸))𝐵)
96, 8sylibr 224 . . . 4 (𝜑𝐴((𝐶𝐷) ∘ 𝐸)𝐵)
104, 5, 9brcoffn 38830 . . 3 (𝜑 → (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵))
111adantr 472 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → 𝐶 Fn 𝑍)
122adantr 472 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → 𝐷:𝑌𝑍)
13 simprr 813 . . . . 5 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → (𝐸𝐴)(𝐶𝐷)𝐵)
1411, 12, 13brcoffn 38830 . . . 4 ((𝜑 ∧ (𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵)) → ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵))
1514ex 449 . . 3 (𝜑 → ((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) → ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)))
1610, 15jcai 560 . 2 (𝜑 → ((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)))
17 simpll 807 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → 𝐴𝐸(𝐸𝐴))
18 simprl 811 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)))
19 simprr 813 . . 3 (((𝐴𝐸(𝐸𝐴) ∧ (𝐸𝐴)(𝐶𝐷)𝐵) ∧ ((𝐸𝐴)𝐷(𝐷‘(𝐸𝐴)) ∧ (𝐷‘(𝐸𝐴))𝐶𝐵)) → (𝐷‘(𝐸𝐴))𝐶𝐵)
2017, 18, 193jca 1123 . 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 383  w3a 1072   class class class wbr 4804  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  brco3f1o  38833  neicvgmex  38917  neicvgel1  38919
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