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Theorem brcoffn 40726
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 6521 . . . 4 ((𝐶 Fn 𝑌𝐷:𝑋𝑌) → (𝐶𝐷) Fn 𝑋)
41, 2, 3syl2anc 587 . . 3 (𝜑 → (𝐶𝐷) Fn 𝑋)
5 simpl 486 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝜑)
6 simpr 488 . . . 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 6434 . . . . 5 (((𝐶𝐷) Fn 𝑋𝐴(𝐶𝐷)𝐵) → 𝐴𝑋)
106, 8, 9syl2anc 587 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝐴𝑋)
115, 6, 103jca 1125 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
124, 11mpdan 686 . 2 (𝜑 → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
1323ad2ant1 1130 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷:𝑋𝑌)
14 simp3 1135 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴𝑋)
15 fvco3 6741 . . . . . 6 ((𝐷:𝑋𝑌𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1613, 14, 15syl2anc 587 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1773ad2ant1 1130 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴(𝐶𝐷)𝐵)
18 fnbrfvb 6697 . . . . . . 7 (((𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
19183adant1 1127 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
2017, 19mpbird 260 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = 𝐵)
2116, 20eqtr3d 2838 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐶‘(𝐷𝐴)) = 𝐵)
22 eqid 2801 . . . 4 (𝐷𝐴) = (𝐷𝐴)
2321, 22jctil 523 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵))
2413ffnd 6492 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷 Fn 𝑋)
25 fnbrfvb 6697 . . . . 5 ((𝐷 Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2624, 14, 25syl2anc 587 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2713ad2ant1 1130 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐶 Fn 𝑌)
2813, 14ffvelrnd 6833 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐷𝐴) ∈ 𝑌)
29 fnbrfvb 6697 . . . . 5 ((𝐶 Fn 𝑌 ∧ (𝐷𝐴) ∈ 𝑌) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3027, 28, 29syl2anc 587 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3126, 30anbi12d 633 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵) ↔ (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵)))
3223, 31mpbid 235 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112   class class class wbr 5033  ccom 5527   Fn wfn 6323  wf 6324  cfv 6328
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336
This theorem is referenced by:  brcofffn  40727  brco2f1o  40728  clsneikex  40802  clsneinex  40803  clsneiel1  40804
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