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Theorem brcoffn 38828
 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 6228 . . . 4 ((𝐶 Fn 𝑌𝐷:𝑋𝑌) → (𝐶𝐷) Fn 𝑋)
41, 2, 3syl2anc 696 . . 3 (𝜑 → (𝐶𝐷) Fn 𝑋)
5 simpl 474 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝜑)
6 simpr 479 . . . 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 6152 . . . . 5 (((𝐶𝐷) Fn 𝑋𝐴(𝐶𝐷)𝐵) → 𝐴𝑋)
106, 8, 9syl2anc 696 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝐴𝑋)
115, 6, 103jca 1123 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
124, 11mpdan 705 . 2 (𝜑 → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
1323ad2ant1 1128 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷:𝑋𝑌)
14 simp3 1133 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴𝑋)
15 fvco3 6435 . . . . . 6 ((𝐷:𝑋𝑌𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1613, 14, 15syl2anc 696 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1773ad2ant1 1128 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴(𝐶𝐷)𝐵)
18 fnbrfvb 6395 . . . . . . 7 (((𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
19183adant1 1125 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
2017, 19mpbird 247 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = 𝐵)
2116, 20eqtr3d 2794 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐶‘(𝐷𝐴)) = 𝐵)
22 eqid 2758 . . . 4 (𝐷𝐴) = (𝐷𝐴)
2321, 22jctil 561 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵))
2413ffnd 6205 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷 Fn 𝑋)
25 fnbrfvb 6395 . . . . 5 ((𝐷 Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2624, 14, 25syl2anc 696 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2713ad2ant1 1128 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐶 Fn 𝑌)
2813, 14ffvelrnd 6521 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐷𝐴) ∈ 𝑌)
29 fnbrfvb 6395 . . . . 5 ((𝐶 Fn 𝑌 ∧ (𝐷𝐴) ∈ 𝑌) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3027, 28, 29syl2anc 696 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3126, 30anbi12d 749 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵) ↔ (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵)))
3223, 31mpbid 222 . 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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1630   ∈ wcel 2137   class class class wbr 4802   ∘ ccom 5268   Fn wfn 6042  ⟶wf 6043  ‘cfv 6047 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fv 6055 This theorem is referenced by:  brcofffn  38829  brco2f1o  38830  clsneikex  38904  clsneinex  38905  clsneiel1  38906
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