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Theorem txcmplem2 21355
Description: Lemma for txcmp 21356. (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 481 . . . . 5 ((𝜑𝑥𝑌) → 𝑅 ∈ Comp)
61adantr 481 . . . . 5 ((𝜑𝑥𝑌) → 𝑆 ∈ Comp)
7 txcmp.w . . . . . 6 (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))
87adantr 481 . . . . 5 ((𝜑𝑥𝑌) → 𝑊 ⊆ (𝑅 ×t 𝑆))
9 txcmp.u . . . . . 6 (𝜑 → (𝑋 × 𝑌) = 𝑊)
109adantr 481 . . . . 5 ((𝜑𝑥𝑌) → (𝑋 × 𝑌) = 𝑊)
11 simpr 477 . . . . 5 ((𝜑𝑥𝑌) → 𝑥𝑌)
122, 3, 5, 6, 8, 10, 11txcmplem1 21354 . . . 4 ((𝜑𝑥𝑌) → ∃𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
1312ralrimiva 2960 . . 3 (𝜑 → ∀𝑥𝑌𝑢𝑆 (𝑥𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))
14 unieq 4410 . . . . 5 (𝑣 = (𝑓𝑢) → 𝑣 = (𝑓𝑢))
1514sseq2d 3612 . . . 4 (𝑣 = (𝑓𝑢) → ((𝑋 × 𝑢) ⊆ 𝑣 ↔ (𝑋 × 𝑢) ⊆ (𝑓𝑢)))
163, 15cmpcovf 21104 . . 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 692 . 2 (𝜑 → ∃𝑤 ∈ (𝒫 𝑆 ∩ Fin)(𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))))
18 simprrl 803 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin))
19 ffn 6002 . . . . . . . . . . 11 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → 𝑓 Fn 𝑤)
20 fniunfv 6459 . . . . . . . . . . 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)
22 frn 6010 . . . . . . . . . . . . 13 (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
2318, 22syl 17 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ (𝒫 𝑊 ∩ Fin))
24 inss1 3811 . . . . . . . . . . . 12 (𝒫 𝑊 ∩ Fin) ⊆ 𝒫 𝑊
2523, 24syl6ss 3595 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓 ⊆ 𝒫 𝑊)
26 sspwuni 4577 . . . . . . . . . . 11 (ran 𝑓 ⊆ 𝒫 𝑊 ran 𝑓𝑊)
2725, 26sylib 208 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ran 𝑓𝑊)
2821, 27eqsstrd 3618 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
29 vex 3189 . . . . . . . . . . 11 𝑤 ∈ V
30 fvex 6158 . . . . . . . . . . 11 (𝑓𝑧) ∈ V
3129, 30iunex 7093 . . . . . . . . . 10 𝑧𝑤 (𝑓𝑧) ∈ V
3231elpw 4136 . . . . . . . . 9 ( 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊 𝑧𝑤 (𝑓𝑧) ⊆ 𝑊)
3328, 32sylibr 224 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ 𝒫 𝑊)
34 inss2 3812 . . . . . . . . . 10 (𝒫 𝑆 ∩ Fin) ⊆ Fin
35 simplr 791 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ (𝒫 𝑆 ∩ Fin))
3634, 35sseldi 3581 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑤 ∈ Fin)
37 inss2 3812 . . . . . . . . . . 11 (𝒫 𝑊 ∩ Fin) ⊆ Fin
38 fss 6013 . . . . . . . . . . 11 ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ (𝒫 𝑊 ∩ Fin) ⊆ Fin) → 𝑓:𝑤⟶Fin)
3918, 37, 38sylancl 693 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑓:𝑤⟶Fin)
40 ffvelrn 6313 . . . . . . . . . . 11 ((𝑓:𝑤⟶Fin ∧ 𝑧𝑤) → (𝑓𝑧) ∈ Fin)
4140ralrimiva 2960 . . . . . . . . . 10 (𝑓:𝑤⟶Fin → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
4239, 41syl 17 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ∈ Fin)
43 iunfi 8198 . . . . . . . . 9 ((𝑤 ∈ Fin ∧ ∀𝑧𝑤 (𝑓𝑧) ∈ Fin) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4436, 42, 43syl2anc 692 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ Fin)
4533, 44elind 3776 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
46 simprl 793 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑤)
47 uniiun 4539 . . . . . . . . . . . . 13 𝑤 = 𝑧𝑤 𝑧
4846, 47syl6eq 2671 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑌 = 𝑧𝑤 𝑧)
4948xpeq2d 5099 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = (𝑋 × 𝑧𝑤 𝑧))
50 xpiundi 5134 . . . . . . . . . . 11 (𝑋 × 𝑧𝑤 𝑧) = 𝑧𝑤 (𝑋 × 𝑧)
5149, 50syl6eq 2671 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑋 × 𝑧))
52 simprrr 804 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))
53 xpeq2 5089 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → (𝑋 × 𝑢) = (𝑋 × 𝑧))
54 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑓𝑢) = (𝑓𝑧))
5554unieqd 4412 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 (𝑓𝑢) = (𝑓𝑧))
5653, 55sseq12d 3613 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → ((𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ (𝑋 × 𝑧) ⊆ (𝑓𝑧)))
5756cbvralv 3159 . . . . . . . . . . . 12 (∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢) ↔ ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
5852, 57sylib 208 . . . . . . . . . . 11 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧))
59 ss2iun 4502 . . . . . . . . . . 11 (∀𝑧𝑤 (𝑋 × 𝑧) ⊆ (𝑓𝑧) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
6058, 59syl 17 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑋 × 𝑧) ⊆ 𝑧𝑤 (𝑓𝑧))
6151, 60eqsstrd 3618 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) ⊆ 𝑧𝑤 (𝑓𝑧))
6218ffvelrnda 6315 . . . . . . . . . . . . . 14 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin))
6324, 62sseldi 3581 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ∈ 𝒫 𝑊)
64 elpwi 4140 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ 𝒫 𝑊 → (𝑓𝑧) ⊆ 𝑊)
65 uniss 4424 . . . . . . . . . . . . 13 ((𝑓𝑧) ⊆ 𝑊 (𝑓𝑧) ⊆ 𝑊)
6663, 64, 653syl 18 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ 𝑊)
679ad3antrrr 765 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑋 × 𝑌) = 𝑊)
6866, 67sseqtr4d 3621 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) ∧ 𝑧𝑤) → (𝑓𝑧) ⊆ (𝑋 × 𝑌))
6968ralrimiva 2960 . . . . . . . . . 10 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
70 iunss 4527 . . . . . . . . . 10 ( 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌) ↔ ∀𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7169, 70sylibr 224 . . . . . . . . 9 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → 𝑧𝑤 (𝑓𝑧) ⊆ (𝑋 × 𝑌))
7261, 71eqssd 3600 . . . . . . . 8 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
73 iuncom4 4494 . . . . . . . 8 𝑧𝑤 (𝑓𝑧) = 𝑧𝑤 (𝑓𝑧)
7472, 73syl6eq 2671 . . . . . . 7 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧))
75 unieq 4410 . . . . . . . . 9 (𝑣 = 𝑧𝑤 (𝑓𝑧) → 𝑣 = 𝑧𝑤 (𝑓𝑧))
7675eqeq2d 2631 . . . . . . . 8 (𝑣 = 𝑧𝑤 (𝑓𝑧) → ((𝑋 × 𝑌) = 𝑣 ↔ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)))
7776rspcev 3295 . . . . . . 7 (( 𝑧𝑤 (𝑓𝑧) ∈ (𝒫 𝑊 ∩ Fin) ∧ (𝑋 × 𝑌) = 𝑧𝑤 (𝑓𝑧)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7845, 74, 77syl2anc 692 . . . . . 6 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ (𝑌 = 𝑤 ∧ (𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
7978expr 642 . . . . 5 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → ((𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8079exlimdv 1858 . . . 4 (((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) ∧ 𝑌 = 𝑤) → (∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢)) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8180expimpd 628 . . 3 ((𝜑𝑤 ∈ (𝒫 𝑆 ∩ Fin)) → ((𝑌 = 𝑤 ∧ ∃𝑓(𝑓:𝑤⟶(𝒫 𝑊 ∩ Fin) ∧ ∀𝑢𝑤 (𝑋 × 𝑢) ⊆ (𝑓𝑢))) → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣))
8281rexlimdva 3024 . 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))
8317, 82mpd 15 1 (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   ciun 4485   × cxp 5072  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Fincfn 7899  Compccmp 21099   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-topgen 16025  df-top 20621  df-bases 20622  df-cmp 21100  df-tx 21275
This theorem is referenced by:  txcmp  21356
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