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Theorem bj-imdirco 36071
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 6251 . . . . . . . 8 ((𝑆𝑅) “ 𝑥) = (𝑆 “ (𝑅𝑥))
21eqeq1i 2738 . . . . . . 7 (((𝑆𝑅) “ 𝑥) = 𝑧 ↔ (𝑆 “ (𝑅𝑥)) = 𝑧)
32anbi2i 624 . . . . . 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 7738 . . . . . . . . . . . 12 (𝜑 → (𝐴 × 𝐵) ∈ V)
8 bj-imdirco.arg1 . . . . . . . . . . . 12 (𝜑𝑅 ⊆ (𝐴 × 𝐵))
97, 8ssexd 5325 . . . . . . . . . . 11 (𝜑𝑅 ∈ V)
10 imaexg 7906 . . . . . . . . . . 11 (𝑅 ∈ V → (𝑅𝑥) ∈ V)
119, 10syl 17 . . . . . . . . . 10 (𝜑 → (𝑅𝑥) ∈ V)
12 imass1 6101 . . . . . . . . . . . . 13 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑥) ⊆ ((𝐴 × 𝐵) “ 𝑥))
13 xpima 6182 . . . . . . . . . . . . . 14 ((𝐴 × 𝐵) “ 𝑥) = if((𝐴𝑥) = ∅, ∅, 𝐵)
14 simpr 486 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑥) = ∅ ∧ 𝑢 ∈ ∅) → 𝑢 ∈ ∅)
15 simpr 486 . . . . . . . . . . . . . . . . . 18 ((¬ (𝐴𝑥) = ∅ ∧ 𝑢𝐵) → 𝑢𝐵)
1614, 15orim12i 908 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑥) = ∅ ∧ 𝑢 ∈ ∅) ∨ (¬ (𝐴𝑥) = ∅ ∧ 𝑢𝐵)) → (𝑢 ∈ ∅ ∨ 𝑢𝐵))
17 elif 4572 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ if((𝐴𝑥) = ∅, ∅, 𝐵) ↔ (((𝐴𝑥) = ∅ ∧ 𝑢 ∈ ∅) ∨ (¬ (𝐴𝑥) = ∅ ∧ 𝑢𝐵)))
18 elun 4149 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ (∅ ∪ 𝐵) ↔ (𝑢 ∈ ∅ ∨ 𝑢𝐵))
1916, 17, 183imtr4i 292 . . . . . . . . . . . . . . . 16 (𝑢 ∈ if((𝐴𝑥) = ∅, ∅, 𝐵) → 𝑢 ∈ (∅ ∪ 𝐵))
2019ssriv 3987 . . . . . . . . . . . . . . 15 if((𝐴𝑥) = ∅, ∅, 𝐵) ⊆ (∅ ∪ 𝐵)
21 0ss 4397 . . . . . . . . . . . . . . . 16 ∅ ⊆ 𝐵
22 ssid 4005 . . . . . . . . . . . . . . . 16 𝐵𝐵
2321, 22unssi 4186 . . . . . . . . . . . . . . 15 (∅ ∪ 𝐵) ⊆ 𝐵
2420, 23sstri 3992 . . . . . . . . . . . . . 14 if((𝐴𝑥) = ∅, ∅, 𝐵) ⊆ 𝐵
2513, 24eqsstri 4017 . . . . . . . . . . . . 13 ((𝐴 × 𝐵) “ 𝑥) ⊆ 𝐵
2612, 25sstrdi 3995 . . . . . . . . . . . 12 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑥) ⊆ 𝐵)
278, 26syl 17 . . . . . . . . . . 11 (𝜑 → (𝑅𝑥) ⊆ 𝐵)
28 eqidd 2734 . . . . . . . . . . 11 (𝜑 → (𝑅𝑥) = (𝑅𝑥))
2927, 28jca 513 . . . . . . . . . 10 (𝜑 → ((𝑅𝑥) ⊆ 𝐵 ∧ (𝑅𝑥) = (𝑅𝑥)))
30 sseq1 4008 . . . . . . . . . . 11 (𝑦 = (𝑅𝑥) → (𝑦𝐵 ↔ (𝑅𝑥) ⊆ 𝐵))
31 eqeq2 2745 . . . . . . . . . . 11 (𝑦 = (𝑅𝑥) → ((𝑅𝑥) = 𝑦 ↔ (𝑅𝑥) = (𝑅𝑥)))
3230, 31anbi12d 632 . . . . . . . . . 10 (𝑦 = (𝑅𝑥) → ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑅𝑥) ⊆ 𝐵 ∧ (𝑅𝑥) = (𝑅𝑥))))
3311, 29, 32spcedv 3589 . . . . . . . . 9 (𝜑 → ∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))
3433biantrurd 534 . . . . . . . 8 (𝜑 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ (∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
35 19.41v 1954 . . . . . . . . 9 (∃𝑦((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ (∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧))
36 anass 470 . . . . . . . . . 10 (((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ (𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
3736exbii 1851 . . . . . . . . 9 (∃𝑦((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
3835, 37bitr3i 277 . . . . . . . 8 ((∃𝑦(𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)))
3934, 38bitrdi 287 . . . . . . 7 (𝜑 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧))))
40 imaeq2 6056 . . . . . . . . . . . . 13 ((𝑅𝑥) = 𝑦 → (𝑆 “ (𝑅𝑥)) = (𝑆𝑦))
4140eqeq1d 2735 . . . . . . . . . . . 12 ((𝑅𝑥) = 𝑦 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ (𝑆𝑦) = 𝑧))
4241pm5.32i 576 . . . . . . . . . . 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 464 . . . . . . . . 9 ((𝑦𝐵 ∧ ((𝑅𝑥) = 𝑦 ∧ (𝑆 “ (𝑅𝑥)) = 𝑧)) ↔ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
4544exbii 1851 . . . . . . . 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 pm4.24 565 . . . . . . . . . . . . 13 (𝑦𝐵 ↔ (𝑦𝐵𝑦𝐵))
4847anbi1i 625 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑦𝐵𝑦𝐵) ∧ (𝑅𝑥) = 𝑦))
49 anass 470 . . . . . . . . . . . 12 (((𝑦𝐵𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ (𝑦𝐵 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
5048, 49bitri 275 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑅𝑥) = 𝑦) ↔ (𝑦𝐵 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
5150anbi2i 624 . . . . . . . . . 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 275 . . . . . . . . 9 (((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
5453exbii 1851 . . . . . . . 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 305 . . . . . 6 (𝜑 → ((𝑆 “ (𝑅𝑥)) = 𝑧 ↔ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))))
5756anbi2d 630 . . . . 5 (𝜑 → (((𝑥𝐴𝑧𝐶) ∧ (𝑆 “ (𝑅𝑥)) = 𝑧) ↔ ((𝑥𝐴𝑧𝐶) ∧ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))))
58 19.42v 1958 . . . . . . 7 (∃𝑦((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ((𝑥𝐴𝑧𝐶) ∧ ∃𝑦(𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))))
59 anass 470 . . . . . . . . 9 (((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (𝑥𝐴 ∧ (𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))))
60 ancom 462 . . . . . . . . . . 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 462 . . . . . . . . . . . 12 (((((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ ((𝑅𝑥) = 𝑦 ∧ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵)))
63 ancom 462 . . . . . . . . . . . . . . 15 ((𝑧𝐶𝑦𝐵) ↔ (𝑦𝐵𝑧𝐶))
6463anbi1i 625 . . . . . . . . . . . . . 14 (((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ↔ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))
6564anbi1i 625 . . . . . . . . . . . . 13 ((((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
66 biid 261 . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . 14 ((((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)) ↔ ((𝑧𝐶𝑦𝐵) ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))
6967, 68bitr4i 278 . . . . . . . . . . . . 13 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (((𝑧𝐶𝑦𝐵) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
70 anass 470 . . . . . . . . . . . . 13 (((((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ↔ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))
7165, 69, 703bitr4i 303 . . . . . . . . . . . 12 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ((((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵) ∧ (𝑅𝑥) = 𝑦))
72 anass 470 . . . . . . . . . . . 12 ((((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵) ↔ ((𝑅𝑥) = 𝑦 ∧ (((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧) ∧ 𝑦𝐵)))
7362, 71, 723bitr4i 303 . . . . . . . . . . 11 ((𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵))
7473anbi2i 624 . . . . . . . . . 10 ((𝑥𝐴 ∧ (𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))) ↔ (𝑥𝐴 ∧ (((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ∧ 𝑦𝐵)))
75 anass 470 . . . . . . . . . 10 ((((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ((𝑅𝑥) = 𝑦 ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
7661, 74, 753bitr4i 303 . . . . . . . . 9 ((𝑥𝐴 ∧ (𝑧𝐶 ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦))))) ↔ (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
7759, 76bitri 275 . . . . . . . 8 (((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ (((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
7877exbii 1851 . . . . . . 7 (∃𝑦((𝑥𝐴𝑧𝐶) ∧ (𝑦𝐵 ∧ ((𝑆𝑦) = 𝑧 ∧ (𝑦𝐵 ∧ (𝑅𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)))
7958, 78bitr3i 277 . . . . . 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 305 . . . 4 (𝜑 → (((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧) ↔ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))))
8281opabbidv 5215 . . 3 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))})
83 bj-opabco 36069 . . 3 ({⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦) ∧ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧))}
8482, 83eqtr4di 2791 . 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 14919 . . . 4 (𝜑 → (𝑆𝑅) ⊆ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)))
88 bj-xpcossxp 36070 . . . 4 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐶)
8987, 88sstrdi 3995 . . 3 (𝜑 → (𝑆𝑅) ⊆ (𝐴 × 𝐶))
905, 85, 89bj-imdirval2 36064 . 2 (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆𝑅)) = {⟨𝑥, 𝑧⟩ ∣ ((𝑥𝐴𝑧𝐶) ∧ ((𝑆𝑅) “ 𝑥) = 𝑧)})
916, 85, 86bj-imdirval2 36064 . . 3 (𝜑 → ((𝐵𝒫*𝐶)‘𝑆) = {⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)})
925, 6, 8bj-imdirval2 36064 . . 3 (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
9391, 92coeq12d 5865 . 2 (𝜑 → (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)) = ({⟨𝑦, 𝑧⟩ ∣ ((𝑦𝐵𝑧𝐶) ∧ (𝑆𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)}))
9484, 90, 933eqtr4d 2783 1 (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cun 3947  cin 3948  wss 3949  c0 4323  ifcif 4529  {copab 5211   × cxp 5675  cima 5680  ccom 5681  cfv 6544  (class class class)co 7409  𝒫*cimdir 36059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-imdir 36060
This theorem is referenced by: (None)
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