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Theorem ptcmplem2 24077
Description: Lemma for ptcmp 24082. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
ptcmp.1 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
ptcmp.2 𝑋 = X𝑛𝐴 (𝐹𝑛)
ptcmp.3 (𝜑𝐴𝑉)
ptcmp.4 (𝜑𝐹:𝐴⟶Comp)
ptcmp.5 (𝜑𝑋 ∈ (UFL ∩ dom card))
ptcmplem2.5 (𝜑𝑈 ⊆ ran 𝑆)
ptcmplem2.6 (𝜑𝑋 = 𝑈)
ptcmplem2.7 (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
Assertion
Ref Expression
ptcmplem2 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
Distinct variable groups:   𝑘,𝑛,𝑢,𝑤,𝑧,𝐴   𝑆,𝑘,𝑛,𝑢,𝑧   𝜑,𝑘,𝑛,𝑢   𝑈,𝑘,𝑢,𝑧   𝑘,𝑉,𝑛,𝑢,𝑤,𝑧   𝑘,𝐹,𝑛,𝑢,𝑤,𝑧   𝑘,𝑋,𝑛,𝑢,𝑤,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑆(𝑤)   𝑈(𝑤,𝑛)

Proof of Theorem ptcmplem2
Dummy variables 𝑓 𝑔 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmplem2.7 . . . 4 (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
2 0ss 4406 . . . . . . 7 ∅ ⊆ 𝑈
3 0fi 9081 . . . . . . 7 ∅ ∈ Fin
4 elfpw 9392 . . . . . . 7 (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ⊆ 𝑈 ∧ ∅ ∈ Fin))
52, 3, 4mpbir2an 711 . . . . . 6 ∅ ∈ (𝒫 𝑈 ∩ Fin)
6 unieq 4923 . . . . . . . 8 (𝑧 = ∅ → 𝑧 = ∅)
7 uni0 4940 . . . . . . . 8 ∅ = ∅
86, 7eqtrdi 2791 . . . . . . 7 (𝑧 = ∅ → 𝑧 = ∅)
98rspceeqv 3645 . . . . . 6 ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
105, 9mpan 690 . . . . 5 (𝑋 = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
1110necon3bi 2965 . . . 4 (¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧𝑋 ≠ ∅)
121, 11syl 17 . . 3 (𝜑𝑋 ≠ ∅)
13 n0 4359 . . 3 (𝑋 ≠ ∅ ↔ ∃𝑓 𝑓𝑋)
1412, 13sylib 218 . 2 (𝜑 → ∃𝑓 𝑓𝑋)
15 ptcmp.2 . . . . . . 7 𝑋 = X𝑛𝐴 (𝐹𝑛)
16 fveq2 6907 . . . . . . . . 9 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1716unieqd 4925 . . . . . . . 8 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1817cbvixpv 8954 . . . . . . 7 X𝑛𝐴 (𝐹𝑛) = X𝑘𝐴 (𝐹𝑘)
1915, 18eqtri 2763 . . . . . 6 𝑋 = X𝑘𝐴 (𝐹𝑘)
20 ptcmp.5 . . . . . . . 8 (𝜑𝑋 ∈ (UFL ∩ dom card))
2120elin2d 4215 . . . . . . 7 (𝜑𝑋 ∈ dom card)
2221adantr 480 . . . . . 6 ((𝜑𝑓𝑋) → 𝑋 ∈ dom card)
2319, 22eqeltrrid 2844 . . . . 5 ((𝜑𝑓𝑋) → X𝑘𝐴 (𝐹𝑘) ∈ dom card)
24 ssrab2 4090 . . . . . 6 {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ⊆ 𝐴
2512adantr 480 . . . . . . 7 ((𝜑𝑓𝑋) → 𝑋 ≠ ∅)
2619, 25eqnetrrid 3014 . . . . . 6 ((𝜑𝑓𝑋) → X𝑘𝐴 (𝐹𝑘) ≠ ∅)
27 eqid 2735 . . . . . . 7 (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o})) = (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}))
2827resixpfo 8975 . . . . . 6 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ⊆ 𝐴X𝑘𝐴 (𝐹𝑘) ≠ ∅) → (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘))
2924, 26, 28sylancr 587 . . . . 5 ((𝜑𝑓𝑋) → (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘))
30 fonum 10096 . . . . 5 ((X𝑘𝐴 (𝐹𝑘) ∈ dom card ∧ (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘)) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
3123, 29, 30syl2anc 584 . . . 4 ((𝜑𝑓𝑋) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
32 vex 3482 . . . . . . . . . . 11 𝑔 ∈ V
33 difexg 5335 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝑓) ∈ V)
3432, 33mp1i 13 . . . . . . . . . 10 ((𝜑𝑓𝑋) → (𝑔𝑓) ∈ V)
35 dmexg 7924 . . . . . . . . . 10 ((𝑔𝑓) ∈ V → dom (𝑔𝑓) ∈ V)
36 uniexg 7759 . . . . . . . . . 10 (dom (𝑔𝑓) ∈ V → dom (𝑔𝑓) ∈ V)
3734, 35, 363syl 18 . . . . . . . . 9 ((𝜑𝑓𝑋) → dom (𝑔𝑓) ∈ V)
3837ralrimivw 3148 . . . . . . . 8 ((𝜑𝑓𝑋) → ∀𝑔𝑋 dom (𝑔𝑓) ∈ V)
39 eqid 2735 . . . . . . . . 9 (𝑔𝑋 dom (𝑔𝑓)) = (𝑔𝑋 dom (𝑔𝑓))
4039fnmpt 6709 . . . . . . . 8 (∀𝑔𝑋 dom (𝑔𝑓) ∈ V → (𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋)
4138, 40syl 17 . . . . . . 7 ((𝜑𝑓𝑋) → (𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋)
42 dffn4 6827 . . . . . . 7 ((𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋 ↔ (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓)))
4341, 42sylib 218 . . . . . 6 ((𝜑𝑓𝑋) → (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓)))
44 fonum 10096 . . . . . 6 ((𝑋 ∈ dom card ∧ (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓))) → ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card)
4522, 43, 44syl2anc 584 . . . . 5 ((𝜑𝑓𝑋) → ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card)
46 ssdif0 4372 . . . . . . . . . . . 12 ( (𝐹𝑘) ⊆ {(𝑓𝑘)} ↔ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅)
47 simpr 484 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) ⊆ {(𝑓𝑘)})
48 simpr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓𝑋) → 𝑓𝑋)
4948, 19eleqtrdi 2849 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝑋) → 𝑓X𝑘𝐴 (𝐹𝑘))
50 vex 3482 . . . . . . . . . . . . . . . . . . . . 21 𝑓 ∈ V
5150elixp 8943 . . . . . . . . . . . . . . . . . . . 20 (𝑓X𝑘𝐴 (𝐹𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘)))
5251simprbi 496 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 (𝐹𝑘) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
5349, 52syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝑋) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
5453r19.21bi 3249 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (𝑓𝑘) ∈ (𝐹𝑘))
5554snssd 4814 . . . . . . . . . . . . . . . 16 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → {(𝑓𝑘)} ⊆ (𝐹𝑘))
5655adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → {(𝑓𝑘)} ⊆ (𝐹𝑘))
5747, 56eqssd 4013 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) = {(𝑓𝑘)})
58 fvex 6920 . . . . . . . . . . . . . . 15 (𝑓𝑘) ∈ V
5958ensn1 9060 . . . . . . . . . . . . . 14 {(𝑓𝑘)} ≈ 1o
6057, 59eqbrtrdi 5187 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) ≈ 1o)
6160ex 412 . . . . . . . . . . . 12 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → ( (𝐹𝑘) ⊆ {(𝑓𝑘)} → (𝐹𝑘) ≈ 1o))
6246, 61biimtrrid 243 . . . . . . . . . . 11 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅ → (𝐹𝑘) ≈ 1o))
6362con3d 152 . . . . . . . . . 10 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1o → ¬ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅))
64 neq0 4358 . . . . . . . . . 10 (¬ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅ ↔ ∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}))
6563, 64imbitrdi 251 . . . . . . . . 9 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1o → ∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})))
66 eldifi 4141 . . . . . . . . . . . . 13 (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → 𝑥 (𝐹𝑘))
67 simplr 769 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → 𝑥 (𝐹𝑘))
68 iftrue 4537 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = 𝑥)
6968, 17eleq12d 2833 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛) ↔ 𝑥 (𝐹𝑘)))
7067, 69syl5ibrcom 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
7148, 15eleqtrdi 2849 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓𝑋) → 𝑓X𝑛𝐴 (𝐹𝑛))
7250elixp 8943 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
7372simprbi 496 . . . . . . . . . . . . . . . . . . . . 21 (𝑓X𝑛𝐴 (𝐹𝑛) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7471, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓𝑋) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7574ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7675r19.21bi 3249 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (𝑓𝑛) ∈ (𝐹𝑛))
77 iffalse 4540 . . . . . . . . . . . . . . . . . . 19 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = (𝑓𝑛))
7877eleq1d 2824 . . . . . . . . . . . . . . . . . 18 𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛) ↔ (𝑓𝑛) ∈ (𝐹𝑛)))
7976, 78syl5ibrcom 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8070, 79pm2.61d 179 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛))
8180ralrimiva 3144 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛))
82 ptcmp.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐴𝑉)
8382ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → 𝐴𝑉)
84 mptelixpg 8974 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8583, 84syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8681, 85mpbird 257 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛))
8786, 15eleqtrrdi 2850 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋)
8866, 87sylan2 593 . . . . . . . . . . . 12 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋)
89 unisnv 4932 . . . . . . . . . . . . 13 {𝑘} = 𝑘
90 simplr 769 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑘𝐴)
91 eleq1w 2822 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝑚𝐴𝑘𝐴))
9290, 91syl5ibrcom 247 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘𝑚𝐴))
9392pm4.71rd 562 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚𝐴𝑚 = 𝑘)))
94 equequ1 2022 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑛 = 𝑘𝑚 = 𝑘))
95 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
9694, 95ifbieq2d 4557 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)))
97 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))
98 vex 3482 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
99 fvex 6920 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓𝑚) ∈ V
10098, 99ifex 4581 . . . . . . . . . . . . . . . . . . . . . . 23 if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ∈ V
10196, 97, 100fvmpt 7016 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚𝐴 → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) = if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)))
102101neeq1d 2998 . . . . . . . . . . . . . . . . . . . . 21 (𝑚𝐴 → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
103102adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
104 iffalse 4540 . . . . . . . . . . . . . . . . . . . . . 22 𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) = (𝑓𝑚))
105104necon1ai 2966 . . . . . . . . . . . . . . . . . . . . 21 (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) → 𝑚 = 𝑘)
106 eldifsni 4795 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → 𝑥 ≠ (𝑓𝑘))
107106ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → 𝑥 ≠ (𝑓𝑘))
108 iftrue 4537 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) = 𝑥)
109 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝑓𝑚) = (𝑓𝑘))
110108, 109neeq12d 3000 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑘 → (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) ↔ 𝑥 ≠ (𝑓𝑘)))
111107, 110syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
112105, 111impbid2 226 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) ↔ 𝑚 = 𝑘))
113103, 112bitrd 279 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ 𝑚 = 𝑘))
114113pm5.32da 579 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → ((𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)) ↔ (𝑚𝐴𝑚 = 𝑘)))
11593, 114bitr4d 282 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))))
116115abbidv 2806 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑚𝑚 = 𝑘} = {𝑚 ∣ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))})
117 df-sn 4632 . . . . . . . . . . . . . . . 16 {𝑘} = {𝑚𝑚 = 𝑘}
118 df-rab 3434 . . . . . . . . . . . . . . . 16 {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)} = {𝑚 ∣ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))}
119116, 117, 1183eqtr4g 2800 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)})
120 fvex 6920 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑛) ∈ V
12198, 120ifex 4581 . . . . . . . . . . . . . . . . . 18 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V
122121rgenw 3063 . . . . . . . . . . . . . . . . 17 𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V
12397fnmpt 6709 . . . . . . . . . . . . . . . . 17 (∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) Fn 𝐴)
124122, 123mp1i 13 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) Fn 𝐴)
125 ixpfn 8942 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑛𝐴 (𝐹𝑛) → 𝑓 Fn 𝐴)
12671, 125syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐴)
127126ad2antrr 726 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑓 Fn 𝐴)
128 fndmdif 7062 . . . . . . . . . . . . . . . 16 (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) Fn 𝐴𝑓 Fn 𝐴) → dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓) = {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)})
129124, 127, 128syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓) = {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)})
130119, 129eqtr4d 2778 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
131130unieqd 4925 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
13289, 131eqtr3id 2789 . . . . . . . . . . . 12 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
133 difeq1 4129 . . . . . . . . . . . . . . 15 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → (𝑔𝑓) = ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
134133dmeqd 5919 . . . . . . . . . . . . . 14 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → dom (𝑔𝑓) = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
135134unieqd 4925 . . . . . . . . . . . . 13 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → dom (𝑔𝑓) = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
136135rspceeqv 3645 . . . . . . . . . . . 12 (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓)) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
13788, 132, 136syl2anc 584 . . . . . . . . . . 11 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
138137ex 412 . . . . . . . . . 10 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
139138exlimdv 1931 . . . . . . . . 9 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
14065, 139syld 47 . . . . . . . 8 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1o → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
141140expimpd 453 . . . . . . 7 ((𝜑𝑓𝑋) → ((𝑘𝐴 ∧ ¬ (𝐹𝑘) ≈ 1o) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
14217breq1d 5158 . . . . . . . . 9 (𝑛 = 𝑘 → ( (𝐹𝑛) ≈ 1o (𝐹𝑘) ≈ 1o))
143142notbid 318 . . . . . . . 8 (𝑛 = 𝑘 → (¬ (𝐹𝑛) ≈ 1o ↔ ¬ (𝐹𝑘) ≈ 1o))
144143elrab 3695 . . . . . . 7 (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ↔ (𝑘𝐴 ∧ ¬ (𝐹𝑘) ≈ 1o))
14539elrnmpt 5972 . . . . . . . 8 (𝑘 ∈ V → (𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓)) ↔ ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
146145elv 3483 . . . . . . 7 (𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓)) ↔ ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
147141, 144, 1463imtr4g 296 . . . . . 6 ((𝜑𝑓𝑋) → (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} → 𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓))))
148147ssrdv 4001 . . . . 5 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ⊆ ran (𝑔𝑋 dom (𝑔𝑓)))
149 ssnum 10077 . . . . 5 ((ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card ∧ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ⊆ ran (𝑔𝑋 dom (𝑔𝑓))) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ dom card)
15045, 148, 149syl2anc 584 . . . 4 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ dom card)
151 xpnum 9989 . . . 4 ((X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card ∧ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ dom card) → (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}) ∈ dom card)
15231, 150, 151syl2anc 584 . . 3 ((𝜑𝑓𝑋) → (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}) ∈ dom card)
15382adantr 480 . . . . 5 ((𝜑𝑓𝑋) → 𝐴𝑉)
154 rabexg 5343 . . . . 5 (𝐴𝑉 → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V)
155153, 154syl 17 . . . 4 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V)
156 fvex 6920 . . . . . . 7 (𝐹𝑘) ∈ V
157156uniex 7760 . . . . . 6 (𝐹𝑘) ∈ V
158157rgenw 3063 . . . . 5 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ V
159 iunexg 7987 . . . . 5 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ V) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ V)
160155, 158, 159sylancl 586 . . . 4 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ V)
161 resixp 8972 . . . . . 6 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ⊆ 𝐴𝑓X𝑘𝐴 (𝐹𝑘)) → (𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘))
16224, 49, 161sylancr 587 . . . . 5 ((𝜑𝑓𝑋) → (𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘))
163162ne0d 4348 . . . 4 ((𝜑𝑓𝑋) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ≠ ∅)
164 ixpiunwdom 9628 . . . 4 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} ∈ V ∧ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ V ∧ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ≠ ∅) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ≼* (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}))
165155, 160, 163, 164syl3anc 1370 . . 3 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ≼* (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}))
166 numwdom 10097 . . 3 (((X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o}) ∈ dom card ∧ 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ≼* (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o})) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
167152, 165, 166syl2anc 584 . 2 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
16814, 167exlimddv 1933 1 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1o} (𝐹𝑘) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  cin 3962  wss 3963  c0 4339  ifcif 4531  𝒫 cpw 4605  {csn 4631   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  cmpo 7433  1oc1o 8498  Xcixp 8936  cen 8981  Fincfn 8984  * cwdom 9602  cardccrd 9973  Compccmp 23410  UFLcufl 23924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-wdom 9603  df-card 9977  df-acn 9980
This theorem is referenced by:  ptcmplem3  24078
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