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Theorem ptcmplem2 22097
Description: Lemma for ptcmp 22102. (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 4127 . . . . . . 7 ∅ ⊆ 𝑈
3 0fin 8365 . . . . . . 7 ∅ ∈ Fin
4 elfpw 8445 . . . . . . 7 (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ⊆ 𝑈 ∧ ∅ ∈ Fin))
52, 3, 4mpbir2an 691 . . . . . 6 ∅ ∈ (𝒫 𝑈 ∩ Fin)
6 unieq 4593 . . . . . . . . 9 (𝑧 = ∅ → 𝑧 = ∅)
7 uni0 4612 . . . . . . . . 9 ∅ = ∅
86, 7syl6eq 2824 . . . . . . . 8 (𝑧 = ∅ → 𝑧 = ∅)
98eqeq2d 2784 . . . . . . 7 (𝑧 = ∅ → (𝑋 = 𝑧𝑋 = ∅))
109rspcev 3465 . . . . . 6 ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
115, 10mpan 671 . . . . 5 (𝑋 = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
1211necon3bi 2972 . . . 4 (¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧𝑋 ≠ ∅)
131, 12syl 17 . . 3 (𝜑𝑋 ≠ ∅)
14 n0 4089 . . 3 (𝑋 ≠ ∅ ↔ ∃𝑓 𝑓𝑋)
1513, 14sylib 209 . 2 (𝜑 → ∃𝑓 𝑓𝑋)
16 ptcmp.2 . . . . . . 7 𝑋 = X𝑛𝐴 (𝐹𝑛)
17 fveq2 6348 . . . . . . . . 9 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1817unieqd 4595 . . . . . . . 8 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1918cbvixpv 8101 . . . . . . 7 X𝑛𝐴 (𝐹𝑛) = X𝑘𝐴 (𝐹𝑘)
2016, 19eqtri 2796 . . . . . 6 𝑋 = X𝑘𝐴 (𝐹𝑘)
21 inss2 3989 . . . . . . . 8 (UFL ∩ dom card) ⊆ dom card
22 ptcmp.5 . . . . . . . 8 (𝜑𝑋 ∈ (UFL ∩ dom card))
2321, 22sseldi 3756 . . . . . . 7 (𝜑𝑋 ∈ dom card)
2423adantr 467 . . . . . 6 ((𝜑𝑓𝑋) → 𝑋 ∈ dom card)
2520, 24syl5eqelr 2858 . . . . 5 ((𝜑𝑓𝑋) → X𝑘𝐴 (𝐹𝑘) ∈ dom card)
26 ssrab2 3843 . . . . . 6 {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ 𝐴
2713adantr 467 . . . . . . 7 ((𝜑𝑓𝑋) → 𝑋 ≠ ∅)
2820, 27syl5eqner 3021 . . . . . 6 ((𝜑𝑓𝑋) → X𝑘𝐴 (𝐹𝑘) ≠ ∅)
29 eqid 2774 . . . . . . 7 (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜})) = (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}))
3029resixpfo 8121 . . . . . 6 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ 𝐴X𝑘𝐴 (𝐹𝑘) ≠ ∅) → (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
3126, 28, 30sylancr 576 . . . . 5 ((𝜑𝑓𝑋) → (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
32 fonum 9102 . . . . 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)
3325, 31, 32syl2anc 574 . . . 4 ((𝜑𝑓𝑋) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
34 vex 3358 . . . . . . . . . . 11 𝑔 ∈ V
35 difexg 4956 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝑓) ∈ V)
3634, 35mp1i 13 . . . . . . . . . 10 ((𝜑𝑓𝑋) → (𝑔𝑓) ∈ V)
37 dmexg 7265 . . . . . . . . . 10 ((𝑔𝑓) ∈ V → dom (𝑔𝑓) ∈ V)
38 uniexg 7123 . . . . . . . . . 10 (dom (𝑔𝑓) ∈ V → dom (𝑔𝑓) ∈ V)
3936, 37, 383syl 18 . . . . . . . . 9 ((𝜑𝑓𝑋) → dom (𝑔𝑓) ∈ V)
4039ralrimivw 3119 . . . . . . . 8 ((𝜑𝑓𝑋) → ∀𝑔𝑋 dom (𝑔𝑓) ∈ V)
41 eqid 2774 . . . . . . . . 9 (𝑔𝑋 dom (𝑔𝑓)) = (𝑔𝑋 dom (𝑔𝑓))
4241fnmpt 6171 . . . . . . . 8 (∀𝑔𝑋 dom (𝑔𝑓) ∈ V → (𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋)
4340, 42syl 17 . . . . . . 7 ((𝜑𝑓𝑋) → (𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋)
44 dffn4 6277 . . . . . . 7 ((𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋 ↔ (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓)))
4543, 44sylib 209 . . . . . 6 ((𝜑𝑓𝑋) → (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓)))
46 fonum 9102 . . . . . 6 ((𝑋 ∈ dom card ∧ (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓))) → ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card)
4724, 45, 46syl2anc 574 . . . . 5 ((𝜑𝑓𝑋) → ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card)
48 ssdif0 4100 . . . . . . . . . . . 12 ( (𝐹𝑘) ⊆ {(𝑓𝑘)} ↔ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅)
49 simpr 472 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) ⊆ {(𝑓𝑘)})
50 simpr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓𝑋) → 𝑓𝑋)
5150, 20syl6eleq 2863 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝑋) → 𝑓X𝑘𝐴 (𝐹𝑘))
52 vex 3358 . . . . . . . . . . . . . . . . . . . . 21 𝑓 ∈ V
5352elixp 8090 . . . . . . . . . . . . . . . . . . . 20 (𝑓X𝑘𝐴 (𝐹𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘)))
5453simprbi 485 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 (𝐹𝑘) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
5551, 54syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝑋) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
5655r19.21bi 3084 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (𝑓𝑘) ∈ (𝐹𝑘))
5756snssd 4486 . . . . . . . . . . . . . . . 16 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → {(𝑓𝑘)} ⊆ (𝐹𝑘))
5857adantr 467 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → {(𝑓𝑘)} ⊆ (𝐹𝑘))
5949, 58eqssd 3775 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) = {(𝑓𝑘)})
60 fvex 6359 . . . . . . . . . . . . . . 15 (𝑓𝑘) ∈ V
6160ensn1 8194 . . . . . . . . . . . . . 14 {(𝑓𝑘)} ≈ 1𝑜
6259, 61syl6eqbr 4836 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) ≈ 1𝑜)
6362ex 398 . . . . . . . . . . . 12 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → ( (𝐹𝑘) ⊆ {(𝑓𝑘)} → (𝐹𝑘) ≈ 1𝑜))
6448, 63syl5bir 234 . . . . . . . . . . 11 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅ → (𝐹𝑘) ≈ 1𝑜))
6564con3d 149 . . . . . . . . . 10 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1𝑜 → ¬ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅))
66 neq0 4088 . . . . . . . . . 10 (¬ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅ ↔ ∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}))
6765, 66syl6ib 242 . . . . . . . . 9 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1𝑜 → ∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})))
68 eldifi 3890 . . . . . . . . . . . . 13 (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → 𝑥 (𝐹𝑘))
69 simplr 774 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → 𝑥 (𝐹𝑘))
70 iftrue 4241 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = 𝑥)
7170, 18eleq12d 2847 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛) ↔ 𝑥 (𝐹𝑘)))
7269, 71syl5ibrcom 238 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
7350, 16syl6eleq 2863 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓𝑋) → 𝑓X𝑛𝐴 (𝐹𝑛))
7452elixp 8090 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
7574simprbi 485 . . . . . . . . . . . . . . . . . . . . 21 (𝑓X𝑛𝐴 (𝐹𝑛) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7673, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓𝑋) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7776ad2antrr 706 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7877r19.21bi 3084 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (𝑓𝑛) ∈ (𝐹𝑛))
79 iffalse 4244 . . . . . . . . . . . . . . . . . . 19 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = (𝑓𝑛))
8079eleq1d 2838 . . . . . . . . . . . . . . . . . 18 𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛) ↔ (𝑓𝑛) ∈ (𝐹𝑛)))
8178, 80syl5ibrcom 238 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8272, 81pm2.61d 171 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛))
8382ralrimiva 3118 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛))
84 ptcmp.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐴𝑉)
8584ad3antrrr 710 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → 𝐴𝑉)
86 mptelixpg 8120 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8785, 86syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛) ↔ ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8883, 87mpbird 248 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛))
8988, 16syl6eleqr 2864 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋)
9068, 89sylan2 581 . . . . . . . . . . . 12 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋)
91 vex 3358 . . . . . . . . . . . . . 14 𝑘 ∈ V
9291unisn 4600 . . . . . . . . . . . . 13 {𝑘} = 𝑘
93 simplr 774 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑘𝐴)
94 eleq1w 2836 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝑚𝐴𝑘𝐴))
9593, 94syl5ibrcom 238 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘𝑚𝐴))
9695pm4.71rd 553 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚𝐴𝑚 = 𝑘)))
97 equequ1 2113 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑛 = 𝑘𝑚 = 𝑘))
98 fveq2 6348 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
9997, 98ifbieq2d 4260 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)))
100 eqid 2774 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))
101 vex 3358 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
102 fvex 6359 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓𝑚) ∈ V
103101, 102ifex 4305 . . . . . . . . . . . . . . . . . . . . . . 23 if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ∈ V
10499, 100, 103fvmpt 6441 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚𝐴 → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) = if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)))
105104neeq1d 3005 . . . . . . . . . . . . . . . . . . . . 21 (𝑚𝐴 → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
106105adantl 468 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
107 iffalse 4244 . . . . . . . . . . . . . . . . . . . . . 22 𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) = (𝑓𝑚))
108107necon1ai 2973 . . . . . . . . . . . . . . . . . . . . 21 (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) → 𝑚 = 𝑘)
109 eldifsni 4468 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → 𝑥 ≠ (𝑓𝑘))
110109ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → 𝑥 ≠ (𝑓𝑘))
111 iftrue 4241 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) = 𝑥)
112 fveq2 6348 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝑓𝑚) = (𝑓𝑘))
113111, 112neeq12d 3007 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑘 → (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) ↔ 𝑥 ≠ (𝑓𝑘)))
114110, 113syl5ibrcom 238 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
115108, 114impbid2 217 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) ↔ 𝑚 = 𝑘))
116106, 115bitrd 269 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ 𝑚 = 𝑘))
117116pm5.32da 569 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → ((𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)) ↔ (𝑚𝐴𝑚 = 𝑘)))
11896, 117bitr4d 272 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))))
119118abbidv 2893 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑚𝑚 = 𝑘} = {𝑚 ∣ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))})
120 df-sn 4327 . . . . . . . . . . . . . . . 16 {𝑘} = {𝑚𝑚 = 𝑘}
121 df-rab 3073 . . . . . . . . . . . . . . . 16 {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)} = {𝑚 ∣ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))}
122119, 120, 1213eqtr4g 2833 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)})
123 fvex 6359 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑛) ∈ V
124101, 123ifex 4305 . . . . . . . . . . . . . . . . . 18 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V
125124rgenw 3076 . . . . . . . . . . . . . . . . 17 𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V
126100fnmpt 6171 . . . . . . . . . . . . . . . . 17 (∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) Fn 𝐴)
127125, 126mp1i 13 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) Fn 𝐴)
128 ixpfn 8089 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑛𝐴 (𝐹𝑛) → 𝑓 Fn 𝐴)
12973, 128syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐴)
130129ad2antrr 706 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑓 Fn 𝐴)
131 fndmdif 6481 . . . . . . . . . . . . . . . 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)})
132127, 130, 131syl2anc 574 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓) = {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)})
133122, 132eqtr4d 2811 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
134133unieqd 4595 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
13592, 134syl5eqr 2822 . . . . . . . . . . . 12 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
136 difeq1 3879 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → (𝑔𝑓) = ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
137136dmeqd 5476 . . . . . . . . . . . . . . 15 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → dom (𝑔𝑓) = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
138137unieqd 4595 . . . . . . . . . . . . . 14 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → dom (𝑔𝑓) = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
139138eqeq2d 2784 . . . . . . . . . . . . 13 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → (𝑘 = dom (𝑔𝑓) ↔ 𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓)))
140139rspcev 3465 . . . . . . . . . . . 12 (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓)) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
14190, 135, 140syl2anc 574 . . . . . . . . . . 11 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
142141ex 398 . . . . . . . . . 10 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
143142exlimdv 2016 . . . . . . . . 9 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
14467, 143syld 47 . . . . . . . 8 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1𝑜 → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
145144expimpd 442 . . . . . . 7 ((𝜑𝑓𝑋) → ((𝑘𝐴 ∧ ¬ (𝐹𝑘) ≈ 1𝑜) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
14618breq1d 4807 . . . . . . . . 9 (𝑛 = 𝑘 → ( (𝐹𝑛) ≈ 1𝑜 (𝐹𝑘) ≈ 1𝑜))
147146notbid 308 . . . . . . . 8 (𝑛 = 𝑘 → (¬ (𝐹𝑛) ≈ 1𝑜 ↔ ¬ (𝐹𝑘) ≈ 1𝑜))
148147elrab 3521 . . . . . . 7 (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ↔ (𝑘𝐴 ∧ ¬ (𝐹𝑘) ≈ 1𝑜))
14941elrnmpt 5522 . . . . . . . 8 (𝑘 ∈ V → (𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓)) ↔ ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
15091, 149ax-mp 5 . . . . . . 7 (𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓)) ↔ ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
151145, 148, 1503imtr4g 286 . . . . . 6 ((𝜑𝑓𝑋) → (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} → 𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓))))
152151ssrdv 3764 . . . . 5 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ ran (𝑔𝑋 dom (𝑔𝑓)))
153 ssnum 9083 . . . . 5 ((ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card ∧ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ ran (𝑔𝑋 dom (𝑔𝑓))) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ dom card)
15447, 152, 153syl2anc 574 . . . 4 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ dom card)
155 xpnum 8998 . . . 4 ((X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card ∧ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ dom card) → (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ dom card)
15633, 154, 155syl2anc 574 . . 3 ((𝜑𝑓𝑋) → (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ dom card)
15784adantr 467 . . . . 5 ((𝜑𝑓𝑋) → 𝐴𝑉)
158 rabexg 4959 . . . . 5 (𝐴𝑉 → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ V)
159157, 158syl 17 . . . 4 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ V)
160 fvex 6359 . . . . . . 7 (𝐹𝑘) ∈ V
161160uniex 7121 . . . . . 6 (𝐹𝑘) ∈ V
162161rgenw 3076 . . . . 5 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V
163 iunexg 7311 . . . . 5 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V)
164159, 162, 163sylancl 575 . . . 4 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V)
165 resixp 8118 . . . . . 6 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ 𝐴𝑓X𝑘𝐴 (𝐹𝑘)) → (𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
16626, 51, 165sylancr 576 . . . . 5 ((𝜑𝑓𝑋) → (𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
167166ne0d 4080 . . . 4 ((𝜑𝑓𝑋) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ≠ ∅)
168 ixpiunwdom 8673 . . . 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}))
169159, 164, 167, 168syl3anc 1480 . . 3 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ≼* (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}))
170 numwdom 9103 . . 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)
171156, 169, 170syl2anc 574 . 2 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
17215, 171exlimddv 2018 1 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 383   = wceq 1634  wex 1855  wcel 2148  {cab 2760  wne 2946  wral 3064  wrex 3065  {crab 3068  Vcvv 3355  cdif 3726  cin 3728  wss 3729  c0 4073  ifcif 4235  𝒫 cpw 4307  {csn 4326   cuni 4585   ciun 4665   class class class wbr 4797  cmpt 4876   × cxp 5261  ccnv 5262  dom cdm 5263  ran crn 5264  cres 5265  cima 5266   Fn wfn 6037  wf 6038  ontowfo 6040  cfv 6042  cmpt2 6814  1𝑜c1o 7727  Xcixp 8083  cen 8127  Fincfn 8130  * cwdom 8639  cardccrd 8982  Compccmp 21430  UFLcufl 21944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-se 5223  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-isom 6051  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-oadd 7738  df-omul 7739  df-er 7917  df-map 8032  df-ixp 8084  df-en 8131  df-dom 8132  df-fin 8134  df-wdom 8641  df-card 8986  df-acn 8989
This theorem is referenced by:  ptcmplem3  22098
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