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Theorem fpwwe2lem2 9707
Description: Lemma for fpwwe2 9718. (Contributed by Mario Carneiro, 19-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
Assertion
Ref Expression
fpwwe2lem2 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,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)

Proof of Theorem fpwwe2lem2
StepHypRef Expression
1 fpwwe2.1 . . . . 5 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
21relopabi 5414 . . . 4 Rel 𝑊
32a1i 11 . . 3 (𝜑 → Rel 𝑊)
4 brrelex12 5324 . . 3 ((Rel 𝑊𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
53, 4sylan 575 . 2 ((𝜑𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6 fpwwe2.2 . . . . 5 (𝜑𝐴 ∈ V)
76adantr 472 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝐴 ∈ V)
8 simprll 797 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋𝐴)
97, 8ssexd 4966 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ∈ V)
10 xpexg 7158 . . . . 5 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
119, 9, 10syl2anc 579 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
12 simprlr 798 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
1311, 12ssexd 4966 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ∈ V)
149, 13jca 507 . 2 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
15 simpl 474 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑥 = 𝑋)
1615sseq1d 3792 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥𝐴𝑋𝐴))
17 simpr 477 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
1815sqxpeqd 5309 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1917, 18sseq12d 3794 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
2016, 19anbi12d 624 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋))))
21 weeq2 5266 . . . . . 6 (𝑥 = 𝑋 → (𝑟 We 𝑥𝑟 We 𝑋))
22 weeq1 5265 . . . . . 6 (𝑟 = 𝑅 → (𝑟 We 𝑋𝑅 We 𝑋))
2321, 22sylan9bb 505 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 We 𝑥𝑅 We 𝑋))
2417cnveqd 5466 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
2524imaeq1d 5647 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 “ {𝑦}) = (𝑅 “ {𝑦}))
2617ineq1d 3975 . . . . . . . . 9 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ (𝑢 × 𝑢)))
2726oveq2d 6858 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))))
2827eqeq1d 2767 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
2925, 28sbceqbid 3603 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → ([(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
3015, 29raleqbidv 3300 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
3123, 30anbi12d 624 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
3220, 31anbi12d 624 . . 3 ((𝑥 = 𝑋𝑟 = 𝑅) → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
3332, 1brabga 5150 . 2 ((𝑋 ∈ V ∧ 𝑅 ∈ V) → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
345, 14, 33pm5.21nd 836 1 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  [wsbc 3596  cin 3731  wss 3732  {csn 4334   class class class wbr 4809  {copab 4871   We wwe 5235   × cxp 5275  ccnv 5276  cima 5280  Rel wrel 5282  (class class class)co 6842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fv 6076  df-ov 6845
This theorem is referenced by:  fpwwe2lem3  9708  fpwwe2lem6  9710  fpwwe2lem7  9711  fpwwe2lem9  9713  fpwwe2lem11  9715  fpwwe2lem12  9716  fpwwe2lem13  9717  fpwwe2  9718  canthwelem  9725  pwfseqlem4  9737
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