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Theorem brcoffn 40258
Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)
Hypotheses
Ref Expression
brcoffn.c (𝜑𝐶 Fn 𝑌)
brcoffn.d (𝜑𝐷:𝑋𝑌)
brcoffn.r (𝜑𝐴(𝐶𝐷)𝐵)
Assertion
Ref Expression
brcoffn (𝜑 → (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵))

Proof of Theorem brcoffn
StepHypRef Expression
1 brcoffn.c . . . 4 (𝜑𝐶 Fn 𝑌)
2 brcoffn.d . . . 4 (𝜑𝐷:𝑋𝑌)
3 fnfco 6536 . . . 4 ((𝐶 Fn 𝑌𝐷:𝑋𝑌) → (𝐶𝐷) Fn 𝑋)
41, 2, 3syl2anc 584 . . 3 (𝜑 → (𝐶𝐷) Fn 𝑋)
5 simpl 483 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝜑)
6 simpr 485 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → (𝐶𝐷) Fn 𝑋)
7 brcoffn.r . . . . . 6 (𝜑𝐴(𝐶𝐷)𝐵)
85, 7syl 17 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝐴(𝐶𝐷)𝐵)
9 fnbr 6452 . . . . 5 (((𝐶𝐷) Fn 𝑋𝐴(𝐶𝐷)𝐵) → 𝐴𝑋)
106, 8, 9syl2anc 584 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝐴𝑋)
115, 6, 103jca 1120 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
124, 11mpdan 683 . 2 (𝜑 → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
1323ad2ant1 1125 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷:𝑋𝑌)
14 simp3 1130 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴𝑋)
15 fvco3 6753 . . . . . 6 ((𝐷:𝑋𝑌𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1613, 14, 15syl2anc 584 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1773ad2ant1 1125 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴(𝐶𝐷)𝐵)
18 fnbrfvb 6711 . . . . . . 7 (((𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
19183adant1 1122 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
2017, 19mpbird 258 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = 𝐵)
2116, 20eqtr3d 2855 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐶‘(𝐷𝐴)) = 𝐵)
22 eqid 2818 . . . 4 (𝐷𝐴) = (𝐷𝐴)
2321, 22jctil 520 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵))
2413ffnd 6508 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷 Fn 𝑋)
25 fnbrfvb 6711 . . . . 5 ((𝐷 Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2624, 14, 25syl2anc 584 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2713ad2ant1 1125 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐶 Fn 𝑌)
2813, 14ffvelrnd 6844 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐷𝐴) ∈ 𝑌)
29 fnbrfvb 6711 . . . . 5 ((𝐶 Fn 𝑌 ∧ (𝐷𝐴) ∈ 𝑌) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3027, 28, 29syl2anc 584 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3126, 30anbi12d 630 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵) ↔ (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵)))
3223, 31mpbid 233 . 2 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵))
3312, 32syl 17 1 (𝜑 → (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   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:  brcofffn  40259  brco2f1o  40260  clsneikex  40334  clsneinex  40335  clsneiel1  40336
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