Theorem bj-imdirid 34478

Description: Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)

Hypothesis

Ref	Expression
bj-imdirid.ex	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
bj-imdirid	⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Proof of Theorem bj-imdirid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-imdirid.ex	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		idssxp 5916	. . . 4 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴))
4	1, 1, 3	bj-imdirval2 34476	. 2 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)})
5		resiima 5944	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
6	5	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴)) → (( I ↾ 𝐴) “ 𝑥) = 𝑥)
7	6	eqeq1d 2823	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴)) → ((( I ↾ 𝐴) “ 𝑥) = 𝑦 ↔ 𝑥 = 𝑦))
8	7	pm5.32da 581	. . . . 5 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = 𝑦)))
9		anass 471	. . . . 5 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑥 = 𝑦) ↔ (𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦)))
10	8, 9	syl6bb 289	. . . 4 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦) ↔ (𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦))))
11		simpr 487	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
12		sseq1 3992	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
13	12	biimpcd 251	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 = 𝑦 → 𝑦 ⊆ 𝐴))
14	13	ancrd 554	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 = 𝑦 → (𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦)))
15	11, 14	impbid2 228	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → ((𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦) ↔ 𝑥 = 𝑦))
16	15	pm5.32i 577	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦))
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦)) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦)))
18		velpw 4544	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
19		vex 3497	. . . . . . 7 ⊢ 𝑦 ∈ V
20	19	ideq 5723	. . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
21	18, 20	anbi12i 628	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = 𝑦))
22	17, 21	syl6bbr 291	. . . 4 ⊢ (𝜑 → ((𝑥 ⊆ 𝐴 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑥 = 𝑦)) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)))
23	10, 22	bitrd 281	. . 3 ⊢ (𝜑 → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)))
24	23	opabbidv 5132	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ (( I ↾ 𝐴) “ 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)})
25		dfres2 5909	. . 3 ⊢ ( I ↾ 𝒫 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)}
26		eqidd 2822	. . 3 ⊢ (𝜑 → ( I ↾ 𝒫 𝐴) = ( I ↾ 𝒫 𝐴))
27	25, 26	syl5eqr 2870	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 I 𝑦)} = ( I ↾ 𝒫 𝐴))
28	4, 24, 27	3eqtrd 2860	1 ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  𝒫 cpw 4539   class class class wbr 5066  {copab 5128   I cid 5459   × cxp 5553   ↾ cres 5557   “ cima 5558  ‘cfv 6355  (class class class)co 7156  𝒫_*cimdir 34473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-imdir 34474

This theorem is referenced by: (None)