Theorem bj-imdirvallem 38102

Description: Lemma for bj-imdirval 38103 and bj-iminvval 38115. (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 5675	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
4	3	pweqd 4573	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5	4	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
6		sseq2 3954	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 ⊆ 𝑎 ↔ 𝑥 ⊆ 𝐴))
7		sseq2 3954	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑦 ⊆ 𝑏 ↔ 𝑦 ⊆ 𝐵))
8	6, 7	bi2anan9 650	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵)))
9	8	anbi1d 643	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)))
10	9	opabbidv 5170	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
11	10	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)})
12	5, 11	mpteq12dv 5191	. 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 3473	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
15		bj-imdirvallem.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
16	15	elexd 3473	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
17	13, 15	xpexd 7765	. . . 4 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
18	17	pwexd 5340	. . 3 ⊢ (𝜑 → 𝒫 (𝐴 × 𝐵) ∈ V)
19	18	mptexd 7230	. 2 ⊢ (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}) ∈ V)
20	2, 12, 14, 16, 19	ovmpod 7572	1 ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3913  𝒫 cpw 4556  {copab 5166   ↦ cmpt 5185   × cxp 5648  (class class class)co 7420   ∈ cmpo 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-dif 3899  df-un 3901  df-in 3903  df-ss 3914  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425

This theorem is used by:  bj-imdirval  38103  bj-iminvval  38115