Theorem bj-iminvval2 37554

Description: Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)

Hypotheses

Ref	Expression
bj-iminvval2.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-iminvval2.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-iminvval2.arg	⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))

Assertion

Ref	Expression
bj-iminvval2	⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 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-iminvval2

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-iminvval2.exa	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		bj-iminvval2.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3	1, 2	bj-iminvval 37553	. 2 ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
4		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
5	4	cnveqd 5817	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅)
6	5	imaeq1d 6011	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (◡𝑟 “ 𝑦) = (◡𝑅 “ 𝑦))
7	6	eqeq2d 2750	. . . 4 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (𝑥 = (◡𝑟 “ 𝑦) ↔ 𝑥 = (◡𝑅 “ 𝑦)))
8	7	anbi2d 636	. . 3 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦)) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))))
9	8	opabbidv 5138	. 2 ⊢ ((𝜑 ∧ 𝑟 = 𝑅) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))})
10	1, 2	xpexd 7694	. . 3 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
11		bj-iminvval2.arg	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))
12	10, 11	sselpwd 5256	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))
13	1, 2	bj-imdirval2lem 37542	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))} ∈ V)
14	3, 9, 12, 13	fvmptd 6943	1 ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529  {copab 5134   × cxp 5616  ^◡ccnv 5617   “ cima 5621  ‘cfv 6485  (class class class)co 7356  𝒫^*ciminv 37551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-iminv 37552

This theorem is referenced by:  bj-iminvid  37555