MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txcmplem2 Structured version   Visualization version   GIF version

Theorem txcmplem2 21854
Description: Lemma for txcmp 21855. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
txcmp.x 𝑋 = 𝑅
txcmp.y 𝑌 = 𝑆
txcmp.r (𝜑𝑅 ∈ Comp)
txcmp.s (𝜑𝑆 ∈ Comp)
txcmp.w (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
txcmp.u (𝜑 → (𝑋 × 𝑌) = 𝑊)
Assertion
Ref Expression
txcmplem2 (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
Distinct variable groups:   𝑣,𝑆   𝑣,𝑌   𝑣,𝑊   𝑣,𝑋
Allowed substitution hints:   𝜑(𝑣)   𝑅(𝑣)

Proof of Theorem txcmplem2
Dummy variables 𝑓 𝑢 𝑥 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txcmp.s . . 3 (𝜑𝑆 ∈ Comp)
2 txcmp.x . . . . 5 𝑋 = 𝑅
3 txcmp.y . . . . 5 𝑌 = 𝑆
4 txcmp.r . . . . . 6 (𝜑𝑅 ∈ Comp)
54adantr 474 . . . . 5 ((𝜑𝑥𝑌) → 𝑅 ∈ Comp)
61adantr 474 . . . . 5 ((𝜑𝑥𝑌) → 𝑆 ∈ Comp)
7 txcmp.w . . . . . 6 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
87adantr 474 . . . . 5 ((𝜑𝑥𝑌) → 𝑊 ⊆ (𝑅 ×t 𝑆))
9 txcmp.u . . . . . 6 (𝜑 → (𝑋 × 𝑌) = 𝑊)
109adantr 474 . . . . 5 ((𝜑𝑥𝑌) → (𝑋 × 𝑌) = 𝑊)
11 simpr 479 . . . . 5 ((𝜑𝑥𝑌) → 𝑥𝑌)
122, 3, 5, 6, 8, 10, 11txcmplem1 21853 . . . 4 ((𝜑𝑥𝑌) → ∃𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
1312ralrimiva 3148 . . 3 (𝜑 → ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
14 unieq 4679 . . . . 5 (𝑣 = (𝑓𝑢) → 𝑣 = (𝑓𝑢))
1514sseq2d 3852 . . . 4 (𝑣 = (𝑓𝑢) → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ (𝑓𝑢)))
163, 15cmpcovf 21603 . . 3 ((𝑆 ∈ Comp ∧ ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣)) → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
171, 13, 16syl2anc 579 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
18 simprrl 771 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin))
19 ffn 6291 . . . . . . . . . . 11 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20 fniunfv 6777 . . . . . . . . . . 11 (𝑓 Fn 𝑤 𝑧𝑤 (𝑓𝑧) = ran 𝑓)
2118, 19, 203syl 18 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) = ran 𝑓)
2218frnd 6298 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
23 inss1 4053 . . . . . . . . . . . 12 (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
2422, 23syl6ss 3833 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
25 sspwuni 4845 . . . . . . . . . . 11 (ran 𝑓 ⊆ 𝒫 𝑊 ran 𝑓𝑊)
2624, 25sylib 210 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓𝑊)
2721, 26eqsstrd 3858 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
28 vex 3401 . . . . . . . . . . 11 𝑤 ∈ V
29 fvex 6459 . . . . . . . . . . 11 (𝑓𝑧) ∈ V
3028, 29iunex 7425 . . . . . . . . . 10 𝑧𝑤 (𝑓𝑧) ∈ V
3130elpw 4385 . . . . . . . . 9 ( 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
3227, 31sylibr 226 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊)
33 inss2 4054 . . . . . . . . . 10 (𝒫 𝑆 ∩ Fin) ⊆ Fin
34 simplr 759 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
3533, 34sseldi 3819 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ Fin)
36 inss2 4054 . . . . . . . . . . 11 (𝒫 𝑊 ∩ Fin) ⊆ Fin
37 fss 6304 . . . . . . . . . . 11 ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
3818, 36, 37sylancl 580 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶Fin)
39 ffvelrn 6621 . . . . . . . . . . 11 ((𝑓:𝑤⟶Fin ∧ 𝑧𝑤) → (𝑓𝑧) ∈ Fin)
4039ralrimiva 3148 . . . . . . . . . 10 (𝑓:𝑤⟶Fin → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
4138, 40syl 17 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
42 iunfi 8542 . . . . . . . . 9 ((𝑤 ∈ Fin ∧ ∀𝑧𝑤 (𝑓𝑧) ∈ Fin) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4335, 41, 42syl2anc 579 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4432, 43elind 4021 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
45 simprl 761 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑤)
46 uniiun 4806 . . . . . . . . . . . . 13 𝑤 = 𝑧𝑤 𝑧
4745, 46syl6eq 2830 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑧𝑤 𝑧)
4847xpeq2d 5385 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = (𝑋 × 𝑧𝑤 𝑧))
49 xpiundi 5419 . . . . . . . . . . 11 (𝑋 × 𝑧𝑤 𝑧) = 𝑧𝑤 (𝑋 × 𝑧)
5048, 49syl6eq 2830 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑋 × 𝑧))
51 simprrr 772 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))
52 xpeq2 5376 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
53 fveq2 6446 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
5453unieqd 4681 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 (𝑓𝑢) = (𝑓𝑧))
5552, 54sseq12d 3853 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ (𝑋 × 𝑧) ⊆ (𝑓𝑧)))
5655cbvralv 3367 . . . . . . . . . . . 12 (∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
5751, 56sylib 210 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
58 ss2iun 4769 . . . . . . . . . . 11 (∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
5957, 58syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
6050, 59eqsstrd 3858 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) ⊆ 𝑧𝑤 (𝑓𝑧))
6118ffvelrnda 6623 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
6223, 61sseldi 3819 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ 𝒫 𝑊)
63 elpwi 4389 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ 𝒫 𝑊 → (𝑓𝑧) ⊆ 𝑊)
64 uniss 4694 . . . . . . . . . . . . 13 ((𝑓𝑧) ⊆ 𝑊 (𝑓𝑧) ⊆ 𝑊)
6562, 63, 643syl 18 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ 𝑊)
669ad3antrrr 720 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑋 × 𝑌) = 𝑊)
6765, 66sseqtr4d 3861 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ (𝑋 × 𝑌))
6867ralrimiva 3148 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
69 iunss 4794 . . . . . . . . . 10 ( 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7068, 69sylibr 226 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7160, 70eqssd 3838 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
72 iuncom4 4761 . . . . . . . 8 𝑧𝑤 (𝑓𝑧) = 𝑧𝑤 (𝑓𝑧)
7371, 72syl6eq 2830 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
74 unieq 4679 . . . . . . . 8 (𝑣 = 𝑧𝑤 (𝑓𝑧) → 𝑣 = 𝑧𝑤 (𝑓𝑧))
7574rspceeqv 3529 . . . . . . 7 (( 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7644, 73, 75syl2anc 579 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7776expr 450 . . . . 5 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
7877exlimdv 1976 . . . 4 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → (∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
7978expimpd 447 . . 3 ((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) → ((𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8079rexlimdva 3213 . 2 (𝜑 → (∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8117, 80mpd 15 1 (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wex 1823  wcel 2107  wral 3090  wrex 3091  cin 3791  wss 3792  𝒫 cpw 4379   cuni 4671   ciun 4753   × cxp 5353  ran crn 5356   Fn wfn 6130  wf 6131  cfv 6135  (class class class)co 6922  Fincfn 8241  Compccmp 21598   ×t ctx 21772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-fin 8245  df-topgen 16490  df-top 21106  df-bases 21158  df-cmp 21599  df-tx 21774
This theorem is referenced by:  txcmp  21855
  Copyright terms: Public domain W3C validator