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Theorem bj-imdirco 34577
Description: Functorial property of the direct image: the direct image by a composition is the composition of the direct images. (Contributed by BJ, 23-May-2024.)
Hypotheses
Ref Expression
bj-imdirco.exa (𝜑𝐴𝑈)
bj-imdirco.exb (𝜑𝐵𝑉)
bj-imdirco.exc (𝜑𝐶𝑊)
bj-imdirco.arg1 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
bj-imdirco.arg2 (𝜑𝑆 ⊆ (𝐵 × 𝐶))
Assertion
Ref Expression
bj-imdirco (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)))

Proof of Theorem bj-imdirco
Dummy variables 𝑥 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imaco 6093 . . . . . . . 8 ((𝑆𝑅) “ 𝑥) = (𝑆 “ (𝑅𝑥))
21eqeq1i 2829 . . . . . . 7 (((𝑆𝑅) “ 𝑥) = 𝑧 ↔ (𝑆 “ (𝑅𝑥)) = 𝑧)
32anbi2i 625 . . . . . 6 (((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧) ↔ ((𝑥𝐴𝑧𝐶) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧))
43a1i 11 . . . . 5 (𝜑 → (((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧) ↔ ((𝑥𝐴𝑧𝐶) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
5 bj-imdirco.exa . . . . . . . . . . . . 13 (𝜑𝐴𝑈)
6 bj-imdirco.exb . . . . . . . . . . . . 13 (𝜑𝐵𝑉)
75, 6xpexd 7470 . . . . . . . . . . . 12 (𝜑 → (𝐴 × 𝐵) ∈ V)
8 bj-imdirco.arg1 . . . . . . . . . . . 12 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
97, 8ssexd 5215 . . . . . . . . . . 11 (𝜑𝑅 ∈ V)
10 imaexg 7617 . . . . . . . . . . 11 (𝑅 ∈ V → (𝑅𝑥) ∈ V)
119, 10syl 17 . . . . . . . . . 10 (𝜑 → (𝑅𝑥) ∈ V)
12 imass1 5953 . . . . . . . . . . . . 13 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑥) ⊆ ((𝐴 × 𝐵) “ 𝑥))
13 xpima 6028 . . . . . . . . . . . . . 14 ((𝐴 × 𝐵) “ 𝑥) = if((𝐴𝑥) = ∅, ∅, 𝐵)
14 simpr 488 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑥) = ∅ ∧ 𝑢 ∈ ∅) → 𝑢 ∈ ∅)
15 simpr 488 . . . . . . . . . . . . . . . . . 18 ((¬ (𝐴𝑥) = ∅ ∧ 𝑢𝐵) → 𝑢𝐵)
1614, 15orim12i 906 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑥) = ∅ ∧ 𝑢 ∈ ∅) ∨ (¬ (𝐴𝑥) = ∅ ∧ 𝑢𝐵)) → (𝑢 ∈ ∅ ∨ 𝑢𝐵))
17 elif 4492 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ if((𝐴𝑥) = ∅, ∅, 𝐵) ↔ (((𝐴𝑥) = ∅ ∧ 𝑢 ∈ ∅) ∨ (¬ (𝐴𝑥) = ∅ ∧ 𝑢𝐵)))
18 elun 4111 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (∅ ∪ 𝐵) ↔ (𝑢 ∈ ∅ ∨ 𝑢𝐵))
1916, 17, 183imtr4i 295 . . . . . . . . . . . . . . . 16 (𝑢 ∈ if((𝐴𝑥) = ∅, ∅, 𝐵) → 𝑢 ∈ (∅ ∪ 𝐵))
2019ssriv 3957 . . . . . . . . . . . . . . 15 if((𝐴𝑥) = ∅, ∅, 𝐵) ⊆ (∅ ∪ 𝐵)
21 0ss 4333 . . . . . . . . . . . . . . . 16 ∅ ⊆ 𝐵
22 ssid 3975 . . . . . . . . . . . . . . . 16 𝐵𝐵
2321, 22unssi 4147 . . . . . . . . . . . . . . 15 (∅ ∪ 𝐵) ⊆ 𝐵
2420, 23sstri 3962 . . . . . . . . . . . . . 14 if((𝐴𝑥) = ∅, ∅, 𝐵) ⊆ 𝐵
2513, 24eqsstri 3987 . . . . . . . . . . . . 13 ((𝐴 × 𝐵) “ 𝑥) ⊆ 𝐵
2612, 25sstrdi 3965 . . . . . . . . . . . 12 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑥) ⊆ 𝐵)
278, 26syl 17 . . . . . . . . . . 11 (𝜑 → (𝑅𝑥) ⊆ 𝐵)
28 eqidd 2825 . . . . . . . . . . 11 (𝜑 → (𝑅𝑥) = (𝑅𝑥))
2927, 28jca 515 . . . . . . . . . 10 (𝜑 → ((𝑅𝑥) ⊆ 𝐵 ∧ (𝑅𝑥) = (𝑅𝑥)))
30 sseq1 3978 . . . . . . . . . . 11 (𝑦 = (𝑅𝑥) → (𝑦𝐵 ↔ (𝑅𝑥) ⊆ 𝐵))
31 eqeq2 2836 . . . . . . . . . . 11 (𝑦 = (𝑅𝑥) → ((𝑅𝑥) = 𝑦 ↔ (𝑅𝑥) = (𝑅𝑥)))
3230, 31anbi12d 633 . . . . . . . . . 10 (𝑦 = (𝑅𝑥) → ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑅𝑥) ⊆ 𝐵 ∧ (𝑅𝑥) = (𝑅𝑥))))
3311, 29, 32spcedv 3585 . . . . . . . . 9 (𝜑 → ∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))
3433biantrurd 536 . . . . . . . 8 (𝜑 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ (∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
35 19.41v 1951 . . . . . . . . 9 (∃𝑦((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ (∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧))
36 anass 472 . . . . . . . . . 10 (((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ (𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
3736exbii 1849 . . . . . . . . 9 (∃𝑦((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
3835, 37bitr3i 280 . . . . . . . 8 ((∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
3934, 38syl6bb 290 . . . . . . 7 (𝜑 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧))))
40 imaeq2 5914 . . . . . . . . . . . . 13 ((𝑅𝑥) = 𝑦 → (𝑆 “ (𝑅𝑥)) = (𝑆𝑦))
4140eqeq1d 2826 . . . . . . . . . . . 12 ((𝑅𝑥) = 𝑦 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ (𝑆𝑦) = 𝑧))
4241pm5.32i 578 . . . . . . . . . . 11 (((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ((𝑅𝑥) = 𝑦 ∧ (𝑆𝑦) = 𝑧))
4342bianass 641 . . . . . . . . . 10 ((𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)) ↔ ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆𝑦) = 𝑧))
4443biancomi 466 . . . . . . . . 9 ((𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)) ↔ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
4544exbii 1849 . . . . . . . 8 (∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)) ↔ ∃𝑦((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
4645a1i 11 . . . . . . 7 (𝜑 → (∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)) ↔ ∃𝑦((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
47 pm4.24 567 . . . . . . . . . . . . 13 (𝑦𝐵 ↔ (𝑦𝐵𝑦𝐵))
4847anbi1i 626 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑦𝐵𝑦𝐵) ∧ (𝑅𝑥) = 𝑦))
49 anass 472 . . . . . . . . . . . 12 (((𝑦𝐵𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ (𝑦𝐵 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
5048, 49bitri 278 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ↔ (𝑦𝐵 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
5150anbi2i 625 . . . . . . . . . 10 (((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
52 an12 644 . . . . . . . . . 10 (((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))) ↔ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
5351, 52bitri 278 . . . . . . . . 9 (((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
5453exbii 1849 . . . . . . . 8 (∃𝑦((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
5554a1i 11 . . . . . . 7 (𝜑 → (∃𝑦((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))))
5639, 46, 553bitrd 308 . . . . . 6 (𝜑 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))))
5756anbi2d 631 . . . . 5 (𝜑 → (((𝑥𝐴𝑧𝐶) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ((𝑥𝐴𝑧𝐶) ∧ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))))
58 19.42v 1955 . . . . . . 7 (∃𝑦((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ((𝑥𝐴𝑧𝐶) ∧ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))))
59 anass 472 . . . . . . . . 9 (((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (𝑥𝐴 ∧ (𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))))
60 ancom 464 . . . . . . . . . . 11 ((((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵) ↔ (𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
6160bianass 641 . . . . . . . . . 10 ((𝑥𝐴 ∧ (((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
62 ancom 464 . . . . . . . . . . . 12 (((((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑅𝑥) = 𝑦 ∧ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵)))
63 ancom 464 . . . . . . . . . . . . . . 15 ((𝑧𝐶𝑦𝐵) ↔ (𝑦𝐵𝑧𝐶))
6463anbi1i 626 . . . . . . . . . . . . . 14 (((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))
6564anbi1i 626 . . . . . . . . . . . . 13 ((((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
66 biid 264 . . . . . . . . . . . . . . 15 ((𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))) ↔ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
6766bianass 641 . . . . . . . . . . . . . 14 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ((𝑧𝐶𝑦𝐵) ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
68 anass 472 . . . . . . . . . . . . . 14 ((((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ ((𝑧𝐶𝑦𝐵) ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
6967, 68bitr4i 281 . . . . . . . . . . . . 13 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
70 anass 472 . . . . . . . . . . . . 13 (((((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
7165, 69, 703bitr4i 306 . . . . . . . . . . . 12 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ((((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵) ∧ (𝑅𝑥) = 𝑦))
72 anass 472 . . . . . . . . . . . 12 ((((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵) ↔ ((𝑅𝑥) = 𝑦 ∧ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵)))
7362, 71, 723bitr4i 306 . . . . . . . . . . 11 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵))
7473anbi2i 625 . . . . . . . . . 10 ((𝑥𝐴 ∧ (𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))) ↔ (𝑥𝐴 ∧ (((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵)))
75 anass 472 . . . . . . . . . 10 ((((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
7661, 74, 753bitr4i 306 . . . . . . . . 9 ((𝑥𝐴 ∧ (𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))) ↔ (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
7759, 76bitri 278 . . . . . . . 8 (((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
7877exbii 1849 . . . . . . 7 (∃𝑦((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
7958, 78bitr3i 280 . . . . . 6 (((𝑥𝐴𝑧𝐶) ∧ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
8079a1i 11 . . . . 5 (𝜑 → (((𝑥𝐴𝑧𝐶) ∧ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
814, 57, 803bitrd 308 . . . 4 (𝜑 → (((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧) ↔ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
8281opabbidv 5119 . . 3 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))})
83 bj-opabco 34575 . . 3 ({⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))}
8482, 83eqtr4di 2877 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧)} = ({⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}))
85 bj-imdirco.exc . . 3 (𝜑𝐶𝑊)
86 bj-imdirco.arg2 . . . . 5 (𝜑𝑆 ⊆ (𝐵 × 𝐶))
8786, 8coss12d 14334 . . . 4 (𝜑 → (𝑆𝑅) ⊆ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)))
88 bj-xpcossxp 34576 . . . 4 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐶)
8987, 88sstrdi 3965 . . 3 (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))
905, 85, 89bj-imdirval2 34570 . 2 (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆𝑅)) = {⟨𝑥, 𝑧⟩ ∣ ((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧)})
916, 85, 86bj-imdirval2 34570 . . 3 (𝜑 → ((𝐵𝒫*𝐶)‘𝑆) = {⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)})
925, 6, 8bj-imdirval2 34570 . . 3 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
9391, 92coeq12d 5723 . 2 (𝜑 → (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)) = ({⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}))
9484, 90, 933eqtr4d 2869 1 (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480  cun 3917  cin 3918  wss 3919  c0 4276  ifcif 4450  {copab 5115   × cxp 5541  cima 5546  ccom 5547  cfv 6345  (class class class)co 7151  𝒫*cimdir 34565
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-imdir 34566
This theorem is referenced by: (None)
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