Theorem bj-imdirval2 37935

Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)

Hypotheses

Ref	Expression
bj-imdirval2.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirval2.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-imdirval2.arg	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))

Assertion

Ref	Expression
bj-imdirval2	⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑈(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem bj-imdirval2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-imdirval2.exa	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		bj-imdirval2.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3	1, 2	bj-imdirval 37933	. 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)}))
4		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	imaeq1d 6055	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑟 “ 𝑥) = (𝑅 “ 𝑥))
6	5	eqeq1d 2762	. . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ((𝑟 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑥) = 𝑦))
7	6	anbi2d 642	. . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)))
8	7	opabbidv 5171	. 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
9	1, 2	xpexd 7750	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
10		bj-imdirval2.arg	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
11	9, 10	sselpwd 5293	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
12	1, 2	bj-imdirval2lem 37934	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ∈ V)
13	3, 8, 11, 12	fvmptd 6994	1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557  {copab 5167   × cxp 5653   “ cima 5658  ‘cfv 6533  (class class class)co 7413  𝒫_*cimdir 37930

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-imdir 37931

This theorem is used by:  bj-imdirval3  37936  bj-imdirid  37938  bj-imdirco  37942