Theorem bj-imdirco 37178

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	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirco.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-imdirco.exc	⊢ (𝜑 → 𝐶 ∈ 𝑊)
bj-imdirco.arg1	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
bj-imdirco.arg2	⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶))

Assertion

Ref	Expression
bj-imdirco	⊢ (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆 ∘ 𝑅)) = (((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.

Step	Hyp	Ref	Expression
1		imaco 6224	. . . . . . . 8 ⊢ ((𝑆 ∘ 𝑅) “ 𝑥) = (𝑆 “ (𝑅 “ 𝑥))
2	1	eqeq1i 2734	. . . . . . 7 ⊢ (((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧 ↔ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)
3	2	anbi2i 623	. . . . . 6 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧))
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)))
5		bj-imdirco.exa	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
6		bj-imdirco.exb	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7	5, 6	xpexd 7727	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
8		bj-imdirco.arg1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
9	7, 8	ssexd 5279	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
10		imaexg 7889	. . . . . . . . . . 11 ⊢ (𝑅 ∈ V → (𝑅 “ 𝑥) ∈ V)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 “ 𝑥) ∈ V)
12		imass1 6072	. . . . . . . . . . . . 13 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 “ 𝑥) ⊆ ((𝐴 × 𝐵) “ 𝑥))
13		xpima 6155	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐵) “ 𝑥) = if((𝐴 ∩ 𝑥) = ∅, ∅, 𝐵)
14		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∩ 𝑥) = ∅ ∧ 𝑢 ∈ ∅) → 𝑢 ∈ ∅)
15		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ (𝐴 ∩ 𝑥) = ∅ ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
16	14, 15	orim12i 908	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∩ 𝑥) = ∅ ∧ 𝑢 ∈ ∅) ∨ (¬ (𝐴 ∩ 𝑥) = ∅ ∧ 𝑢 ∈ 𝐵)) → (𝑢 ∈ ∅ ∨ 𝑢 ∈ 𝐵))
17		elif 4532	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ if((𝐴 ∩ 𝑥) = ∅, ∅, 𝐵) ↔ (((𝐴 ∩ 𝑥) = ∅ ∧ 𝑢 ∈ ∅) ∨ (¬ (𝐴 ∩ 𝑥) = ∅ ∧ 𝑢 ∈ 𝐵)))
18		elun 4116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (∅ ∪ 𝐵) ↔ (𝑢 ∈ ∅ ∨ 𝑢 ∈ 𝐵))
19	16, 17, 18	3imtr4i 292	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ if((𝐴 ∩ 𝑥) = ∅, ∅, 𝐵) → 𝑢 ∈ (∅ ∪ 𝐵))
20	19	ssriv 3950	. . . . . . . . . . . . . . 15 ⊢ if((𝐴 ∩ 𝑥) = ∅, ∅, 𝐵) ⊆ (∅ ∪ 𝐵)
21		0ss 4363	. . . . . . . . . . . . . . . 16 ⊢ ∅ ⊆ 𝐵
22		ssid 3969	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ⊆ 𝐵
23	21, 22	unssi 4154	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ 𝐵) ⊆ 𝐵
24	20, 23	sstri 3956	. . . . . . . . . . . . . 14 ⊢ if((𝐴 ∩ 𝑥) = ∅, ∅, 𝐵) ⊆ 𝐵
25	13, 24	eqsstri 3993	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) “ 𝑥) ⊆ 𝐵
26	12, 25	sstrdi 3959	. . . . . . . . . . . 12 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 “ 𝑥) ⊆ 𝐵)
27	8, 26	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 “ 𝑥) ⊆ 𝐵)
28		eqidd 2730	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 “ 𝑥) = (𝑅 “ 𝑥))
29	27, 28	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑅 “ 𝑥) ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = (𝑅 “ 𝑥)))
30		sseq1 3972	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑅 “ 𝑥) → (𝑦 ⊆ 𝐵 ↔ (𝑅 “ 𝑥) ⊆ 𝐵))
31		eqeq2 2741	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑅 “ 𝑥) → ((𝑅 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑥) = (𝑅 “ 𝑥)))
32	30, 31	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = (𝑅 “ 𝑥) → ((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ↔ ((𝑅 “ 𝑥) ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = (𝑅 “ 𝑥))))
33	11, 29, 32	spcedv 3564	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑦(𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))
34	33	biantrurd 532	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 “ (𝑅 “ 𝑥)) = 𝑧 ↔ (∃𝑦(𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)))
35		19.41v 1949	. . . . . . . . 9 ⊢ (∃𝑦((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧) ↔ (∃𝑦(𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧))
36		anass 468	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧) ↔ (𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)))
37	36	exbii 1848	. . . . . . . . 9 ⊢ (∃𝑦((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)))
38	35, 37	bitr3i 277	. . . . . . . 8 ⊢ ((∃𝑦(𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)))
39	34, 38	bitrdi 287	. . . . . . 7 ⊢ (𝜑 → ((𝑆 “ (𝑅 “ 𝑥)) = 𝑧 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧))))
40		imaeq2 6027	. . . . . . . . . . . . 13 ⊢ ((𝑅 “ 𝑥) = 𝑦 → (𝑆 “ (𝑅 “ 𝑥)) = (𝑆 “ 𝑦))
41	40	eqeq1d 2731	. . . . . . . . . . . 12 ⊢ ((𝑅 “ 𝑥) = 𝑦 → ((𝑆 “ (𝑅 “ 𝑥)) = 𝑧 ↔ (𝑆 “ 𝑦) = 𝑧))
42	41	pm5.32i 574	. . . . . . . . . . 11 ⊢ (((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧) ↔ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ 𝑦) = 𝑧))
43	42	bianass 642	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)) ↔ ((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ∧ (𝑆 “ 𝑦) = 𝑧))
44	43	biancomi 462	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)) ↔ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
45	44	exbii 1848	. . . . . . . 8 ⊢ (∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)) ↔ ∃𝑦((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
46	45	a1i 11	. . . . . . 7 ⊢ (𝜑 → (∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧)) ↔ ∃𝑦((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
47		pm4.24 563	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ 𝐵 ↔ (𝑦 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵))
48	47	anbi1i 624	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ↔ ((𝑦 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦))
49		anass 468	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ↔ (𝑦 ⊆ 𝐵 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
50	48, 49	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦) ↔ (𝑦 ⊆ 𝐵 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
51	50	anbi2i 623	. . . . . . . . . 10 ⊢ (((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)) ↔ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
52		an12 645	. . . . . . . . . 10 ⊢ (((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))) ↔ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
53	51, 52	bitri 275	. . . . . . . . 9 ⊢ (((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)) ↔ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
54	53	exbii 1848	. . . . . . . 8 ⊢ (∃𝑦((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
55	54	a1i 11	. . . . . . 7 ⊢ (𝜑 → (∃𝑦((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))))
56	39, 46, 55	3bitrd 305	. . . . . 6 ⊢ (𝜑 → ((𝑆 “ (𝑅 “ 𝑥)) = 𝑧 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))))
57	56	anbi2d 630	. . . . 5 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ (𝑅 “ 𝑥)) = 𝑧) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))))
58		19.42v 1953	. . . . . . 7 ⊢ (∃𝑦((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))))
59		anass 468	. . . . . . . . 9 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ (𝑥 ⊆ 𝐴 ∧ (𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))))
60		ancom 460	. . . . . . . . . . 11 ⊢ ((((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)) ∧ 𝑦 ⊆ 𝐵) ↔ (𝑦 ⊆ 𝐵 ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))))
61	60	bianass 642	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ (((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)) ∧ 𝑦 ⊆ 𝐵)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))))
62		ancom 460	. . . . . . . . . . . 12 ⊢ (((((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ↔ ((𝑅 “ 𝑥) = 𝑦 ∧ (((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ 𝑦 ⊆ 𝐵)))
63		ancom 460	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶))
64	63	anbi1i 624	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑆 “ 𝑦) = 𝑧) ↔ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))
65	64	anbi1i 624	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)) ↔ (((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
66		biid 261	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))) ↔ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
67	66	bianass 642	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ ((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
68		anass 468	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)) ↔ ((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))
69	67, 68	bitr4i 278	. . . . . . . . . . . . 13 ⊢ ((𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ (((𝑧 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
70		anass 468	. . . . . . . . . . . . 13 ⊢ (((((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ↔ (((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))
71	65, 69, 70	3bitr4i 303	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ ((((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦))
72		anass 468	. . . . . . . . . . . 12 ⊢ ((((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)) ∧ 𝑦 ⊆ 𝐵) ↔ ((𝑅 “ 𝑥) = 𝑦 ∧ (((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧) ∧ 𝑦 ⊆ 𝐵)))
73	62, 71, 72	3bitr4i 303	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ (((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)) ∧ 𝑦 ⊆ 𝐵))
74	73	anbi2i 623	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))) ↔ (𝑥 ⊆ 𝐴 ∧ (((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)) ∧ 𝑦 ⊆ 𝐵)))
75		anass 468	. . . . . . . . . 10 ⊢ ((((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ ((𝑅 “ 𝑥) = 𝑦 ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))))
76	61, 74, 75	3bitr4i 303	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑧 ⊆ 𝐶 ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦))))) ↔ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)))
77	59, 76	bitri 275	. . . . . . . 8 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)))
78	77	exbii 1848	. . . . . . 7 ⊢ (∃𝑦((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)))
79	58, 78	bitr3i 277	. . . . . 6 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)))
80	79	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ∃𝑦(𝑦 ⊆ 𝐵 ∧ ((𝑆 “ 𝑦) = 𝑧 ∧ (𝑦 ⊆ 𝐵 ∧ (𝑅 “ 𝑥) = 𝑦)))) ↔ ∃𝑦(((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))))
81	4, 57, 80	3bitrd 305	. . . 4 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧) ↔ ∃𝑦(((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))))
82	81	opabbidv 5173	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))})
83		bj-opabco 37176	. . 3 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦) ∧ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧))}
84	82, 83	eqtr4di 2782	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧)} = ({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}))
85		bj-imdirco.exc	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
86		bj-imdirco.arg2	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶))
87	86, 8	coss12d 14938	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)))
88		bj-xpcossxp 37177	. . . 4 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐶)
89	87, 88	sstrdi 3959	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑅) ⊆ (𝐴 × 𝐶))
90	5, 85, 89	bj-imdirval2 37171	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆 ∘ 𝑅)) = {⟨𝑥, 𝑧⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐶) ∧ ((𝑆 ∘ 𝑅) “ 𝑥) = 𝑧)})
91	6, 85, 86	bj-imdirval2 37171	. . 3 ⊢ (𝜑 → ((𝐵𝒫*𝐶)‘𝑆) = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)})
92	5, 6, 8	bj-imdirval2 37171	. . 3 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
93	91, 92	coeq12d 5828	. 2 ⊢ (𝜑 → (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)) = ({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝐶) ∧ (𝑆 “ 𝑦) = 𝑧)} ∘ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)}))
94	84, 90, 93	3eqtr4d 2774	1 ⊢ (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆 ∘ 𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ifcif 4488  {copab 5169   × cxp 5636   “ cima 5641   ∘ ccom 5642  ‘cfv 6511  (class class class)co 7387  𝒫_*cimdir 37166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-imdir 37167

This theorem is referenced by: (None)