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Theorem brcoffn 37348
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 5982 . . . 4 ((𝐶 Fn 𝑌𝐷:𝑋𝑌) → (𝐶𝐷) Fn 𝑋)
41, 2, 3syl2anc 691 . . 3 (𝜑 → (𝐶𝐷) Fn 𝑋)
5 simpl 472 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝜑)
6 simpr 476 . . . 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 5907 . . . . 5 (((𝐶𝐷) Fn 𝑋𝐴(𝐶𝐷)𝐵) → 𝐴𝑋)
106, 8, 9syl2anc 691 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → 𝐴𝑋)
115, 6, 103jca 1235 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋) → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
124, 11mpdan 699 . 2 (𝜑 → (𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋))
1323ad2ant1 1075 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷:𝑋𝑌)
14 simp3 1056 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴𝑋)
15 fvco3 6185 . . . . . 6 ((𝐷:𝑋𝑌𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1613, 14, 15syl2anc 691 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = (𝐶‘(𝐷𝐴)))
1773ad2ant1 1075 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐴(𝐶𝐷)𝐵)
18 fnbrfvb 6146 . . . . . . 7 (((𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
19183adant1 1072 . . . . . 6 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐶𝐷)‘𝐴) = 𝐵𝐴(𝐶𝐷)𝐵))
2017, 19mpbird 246 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶𝐷)‘𝐴) = 𝐵)
2116, 20eqtr3d 2646 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐶‘(𝐷𝐴)) = 𝐵)
22 eqid 2610 . . . 4 (𝐷𝐴) = (𝐷𝐴)
2321, 22jctil 558 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵))
2413ffnd 5959 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐷 Fn 𝑋)
25 fnbrfvb 6146 . . . . 5 ((𝐷 Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2624, 14, 25syl2anc 691 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐷𝐴) = (𝐷𝐴) ↔ 𝐴𝐷(𝐷𝐴)))
2713ad2ant1 1075 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → 𝐶 Fn 𝑌)
2813, 14ffvelrnd 6268 . . . . 5 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (𝐷𝐴) ∈ 𝑌)
29 fnbrfvb 6146 . . . . 5 ((𝐶 Fn 𝑌 ∧ (𝐷𝐴) ∈ 𝑌) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3027, 28, 29syl2anc 691 . . . 4 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → ((𝐶‘(𝐷𝐴)) = 𝐵 ↔ (𝐷𝐴)𝐶𝐵))
3126, 30anbi12d 743 . . 3 ((𝜑 ∧ (𝐶𝐷) Fn 𝑋𝐴𝑋) → (((𝐷𝐴) = (𝐷𝐴) ∧ (𝐶‘(𝐷𝐴)) = 𝐵) ↔ (𝐴𝐷(𝐷𝐴) ∧ (𝐷𝐴)𝐶𝐵)))
3223, 31mpbid 221 . 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583  ccom 5042   Fn wfn 5799  wf 5800  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812
This theorem is referenced by:  brcofffn  37349  brco2f1o  37350  clsneikex  37424  clsneinex  37425  clsneiel1  37426
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