Theorem bj-imdirvallem 34860

Description: Lemma for bj-imdirval 34861 and bj-iminvval 34873. (Contributed by BJ, 23-May-2024.)

Hypotheses

Ref	Expression
bj-imdirvallem.1	⊢ (𝜑 → 𝐴 ∈ 𝑈)
bj-imdirvallem.2	⊢ (𝜑 → 𝐵 ∈ 𝑉)
bj-imdirvallem.df	⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}))

Assertion

Ref	Expression
bj-imdirvallem	⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥,𝑦   𝐵,𝑎,𝑏,𝑟,𝑥,𝑦   𝜑,𝑎,𝑏,𝑟   𝜓,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑟)   𝐶(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑈(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem bj-imdirvallem

Step	Hyp	Ref	Expression
1		bj-imdirvallem.df	. . 3 ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}))
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)})))
3		xpeq12 5542	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
4	3	pweqd 4506	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5	4	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
6		sseq2 3914	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝑥 ⊆ 𝐴))
7		sseq2 3914	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑦 ⊆ 𝑏 ↔ 𝑦 ⊆ 𝐵))
8	6, 7	bi2anan9 639	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)))
9	8	anbi1d 633	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)))
10	9	opabbidv 5091	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
11	10	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
12	5, 11	mpteq12dv 5110	. 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))
13		bj-imdirvallem.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
14	13	elexd 3429	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
15		bj-imdirvallem.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
16	15	elexd 3429	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
17	13, 15	xpexd 7465	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
18	17	pwexd 5241	. . 3 ⊢ (𝜑 → 𝒫 (𝐴 × 𝐵) ∈ V)
19	18	mptexd 6971	. 2 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}) ∈ V)
20	2, 12, 14, 16, 19	ovmpod 7290	1 ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3407   ⊆ wss 3854  𝒫 cpw 4487  {copab 5087   ↦ cmpt 5105   × cxp 5515  (class class class)co 7143   ∈ cmpo 7145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148

This theorem is referenced by:  bj-imdirval  34861  bj-iminvval  34873