Theorem fpwwe2lem3 9455

Description: Lemma for fpwwe2 9465. (Contributed by Mario Carneiro, 19-May-2015.)

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)
fpwwe2lem4.4	⊢ (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)

Proof of Theorem fpwwe2lem3

Step	Hyp	Ref	Expression
1		fpwwe2lem4.4	. . . . 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 9454	. . . . 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 222	. . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
6	5	simprrd 797	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)
7		sneq 4187	. . . . . 6 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
8	7	imaeq2d 5466	. . . . 5 ⊢ (𝑦 = 𝐵 → (◡𝑅 “ {𝑦}) = (◡𝑅 “ {𝐵}))
9		eqeq2 2633	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
10	8, 9	sbceqbid 3442	. . . 4 ⊢ (𝑦 = 𝐵 → ([(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
11	10	rspccva 3308	. . 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 488	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → [(◡𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
13		cnvimass 5485	. . . . 5 ⊢ (◡𝑅 “ {𝐵}) ⊆ dom 𝑅
14	2	relopabi 5245	. . . . . . 7 ⊢ Rel 𝑊
15	14	brrelex2i 5159	. . . . . 6 ⊢ (𝑋𝑊𝑅 → 𝑅 ∈ V)
16		dmexg 7097	. . . . . 6 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
17	1, 15, 16	3syl 18	. . . . 5 ⊢ (𝜑 → dom 𝑅 ∈ V)
18		ssexg 4804	. . . . 5 ⊢ (((◡𝑅 “ {𝐵}) ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → (◡𝑅 “ {𝐵}) ∈ V)
19	13, 17, 18	sylancr 695	. . . 4 ⊢ (𝜑 → (◡𝑅 “ {𝐵}) ∈ V)
20		id 22	. . . . . . 7 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → 𝑢 = (◡𝑅 “ {𝐵}))
21	20	sqxpeqd 5141	. . . . . . . 8 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → (𝑢 × 𝑢) = ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))
22	21	ineq2d 3814	. . . . . . 7 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → (𝑅 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵}))))
23	20, 22	oveq12d 6668	. . . . . 6 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))))
24	23	eqeq1d 2624	. . . . 5 ⊢ (𝑢 = (◡𝑅 “ {𝐵}) → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵))
25	24	sbcieg 3468	. . . 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 481	. 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 222	1 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435   ∩ cin 3573   ⊆ wss 3574  {csn 4177   class class class wbr 4653  {copab 4712   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  fpwwe2lem8  9459  fpwwe2lem12  9463  fpwwe2lem13  9464  fpwwe2  9465  canthwelem  9472  pwfseqlem4  9484