Theorem fpwwe2lem3 10544

Description: Lemma for fpwwe2 10554. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe2.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe2lem3.4	⊢ (𝜑 → 𝑋𝑊𝑅)

Assertion

Ref	Expression
fpwwe2lem3	⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵)

Distinct variable groups:   𝑦,𝑢,𝐵   𝑢,𝑟,𝑥,𝑦,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑢)   𝐵(𝑥,𝑟)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem3

Step	Hyp	Ref	Expression
1		fpwwe2lem3.4	. . . . 5 ⊢ (𝜑 → 𝑋𝑊𝑅)
2		fpwwe2.1	. . . . . 6 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3		fpwwe2.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4	2, 3	fpwwe2lem2 10543	. . . . 5 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
5	1, 4	mpbid 232	. . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
6	5	simprrd 773	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)
7		sneq 4590	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
8	7	imaeq2d 6019	. . . . 5 ⊢ (𝑦 = 𝐵 → (◡𝑅 “ {𝑦}) = (◡𝑅 “ {𝐵}))
9		eqeq2 2748	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
10	8, 9	sbceqbid 3747	. . . 4 ⊢ (𝑦 = 𝐵 → ([(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
11	10	rspccva 3575	. . 3 ⊢ ((∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ∧ 𝐵 ∈ 𝑋) → [(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
12	6, 11	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → [(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
13		cnvimass 6041	. . . . 5 ⊢ (◡𝑅 “ {𝐵}) ⊆ dom 𝑅
14	2	relopabiv 5769	. . . . . . 7 ⊢ Rel 𝑊
15	14	brrelex2i 5681	. . . . . 6 ⊢ (𝑋𝑊𝑅 → 𝑅 ∈ V)
16		dmexg 7843	. . . . . 6 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
17	1, 15, 16	3syl 18	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
18		ssexg 5268	. . . . 5 ⊢ (((◡𝑅 “ {𝐵}) ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → (◡𝑅 “ {𝐵}) ∈ V)
19	13, 17, 18	sylancr 587	. . . 4 ⊢ (𝜑 → (◡𝑅 “ {𝐵}) ∈ V)
20		id 22	. . . . . . 7 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → 𝑢 = (◡𝑅 “ {𝐵}))
21	20	sqxpeqd 5656	. . . . . . . 8 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → (𝑢 × 𝑢) = ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))
22	21	ineq2d 4172	. . . . . . 7 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → (𝑅 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵}))))
23	20, 22	oveq12d 7376	. . . . . 6 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))))
24	23	eqeq1d 2738	. . . . 5 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵))
25	24	sbcieg 3780	. . . 4 ⊢ ((◡𝑅 “ {𝐵}) ∈ V → ([(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵))
26	19, 25	syl 17	. . 3 ⊢ (𝜑 → ([(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵))
27	26	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → ([(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵))
28	12, 27	mpbid 232	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440  [wsbc 3740   ∩ cin 3900   ⊆ wss 3901  {csn 4580   class class class wbr 5098  {copab 5160   We wwe 5576   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   “ cima 5627  (class class class)co 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  fpwwe2lem7  10548  fpwwe2lem11  10552  fpwwe2lem12  10553  fpwwe2  10554  canthwelem  10561  pwfseqlem4  10573