Theorem bj-imdirval2 34598

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 34596	. 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)}))
4		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	imaeq1d 5895	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑟 “ 𝑥) = (𝑅 “ 𝑥))
6	5	eqeq1d 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ((𝑟 “ 𝑥) = 𝑦 ↔ (𝑅 “ 𝑥) = 𝑦))
7	6	anbi2d 631	. . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)))
8	7	opabbidv 5096	. 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 7454	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
10		bj-imdirval2.arg	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
11	9, 10	sselpwd 5194	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
12	1, 2	bj-imdirval2lem 34597	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)} ∈ V)
13	3, 8, 11, 12	fvmptd 6752	1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  {copab 5092   × cxp 5517   “ cima 5522  ‘cfv 6324  (class class class)co 7135  𝒫_*cimdir 34593

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-imdir 34594

This theorem is referenced by:  bj-imdirval3  34599  bj-imdirid  34601  bj-imdirco  34605