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Theorem ptcmplem2 21904
Description: Lemma for ptcmp 21909. (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 4005 . . . . . . 7 ∅ ⊆ 𝑈
3 0fin 8229 . . . . . . 7 ∅ ∈ Fin
4 elfpw 8309 . . . . . . 7 (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ⊆ 𝑈 ∧ ∅ ∈ Fin))
52, 3, 4mpbir2an 975 . . . . . 6 ∅ ∈ (𝒫 𝑈 ∩ Fin)
6 unieq 4476 . . . . . . . . 9 (𝑧 = ∅ → 𝑧 = ∅)
7 uni0 4497 . . . . . . . . 9 ∅ = ∅
86, 7syl6eq 2701 . . . . . . . 8 (𝑧 = ∅ → 𝑧 = ∅)
98eqeq2d 2661 . . . . . . 7 (𝑧 = ∅ → (𝑋 = 𝑧𝑋 = ∅))
109rspcev 3340 . . . . . 6 ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
115, 10mpan 706 . . . . 5 (𝑋 = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)
1211necon3bi 2849 . . . 4 (¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧𝑋 ≠ ∅)
131, 12syl 17 . . 3 (𝜑𝑋 ≠ ∅)
14 n0 3964 . . 3 (𝑋 ≠ ∅ ↔ ∃𝑓 𝑓𝑋)
1513, 14sylib 208 . 2 (𝜑 → ∃𝑓 𝑓𝑋)
16 ptcmp.2 . . . . . . 7 𝑋 = X𝑛𝐴 (𝐹𝑛)
17 fveq2 6229 . . . . . . . . 9 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
1817unieqd 4478 . . . . . . . 8 (𝑛 = 𝑘 (𝐹𝑛) = (𝐹𝑘))
1918cbvixpv 7968 . . . . . . 7 X𝑛𝐴 (𝐹𝑛) = X𝑘𝐴 (𝐹𝑘)
2016, 19eqtri 2673 . . . . . 6 𝑋 = X𝑘𝐴 (𝐹𝑘)
21 inss2 3867 . . . . . . . 8 (UFL ∩ dom card) ⊆ dom card
22 ptcmp.5 . . . . . . . 8 (𝜑𝑋 ∈ (UFL ∩ dom card))
2321, 22sseldi 3634 . . . . . . 7 (𝜑𝑋 ∈ dom card)
2423adantr 480 . . . . . 6 ((𝜑𝑓𝑋) → 𝑋 ∈ dom card)
2520, 24syl5eqelr 2735 . . . . 5 ((𝜑𝑓𝑋) → X𝑘𝐴 (𝐹𝑘) ∈ dom card)
26 ssrab2 3720 . . . . . 6 {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ 𝐴
2713adantr 480 . . . . . . 7 ((𝜑𝑓𝑋) → 𝑋 ≠ ∅)
2820, 27syl5eqner 2898 . . . . . 6 ((𝜑𝑓𝑋) → X𝑘𝐴 (𝐹𝑘) ≠ ∅)
29 eqid 2651 . . . . . . 7 (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜})) = (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}))
3029resixpfo 7988 . . . . . 6 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ 𝐴X𝑘𝐴 (𝐹𝑘) ≠ ∅) → (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
3126, 28, 30sylancr 696 . . . . 5 ((𝜑𝑓𝑋) → (𝑔X𝑘𝐴 (𝐹𝑘) ↦ (𝑔 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜})):X𝑘𝐴 (𝐹𝑘)–ontoX𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
32 fonum 8919 . . . . 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 694 . . . 4 ((𝜑𝑓𝑋) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
34 vex 3234 . . . . . . . . . . 11 𝑔 ∈ V
35 difexg 4841 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝑓) ∈ V)
3634, 35mp1i 13 . . . . . . . . . 10 ((𝜑𝑓𝑋) → (𝑔𝑓) ∈ V)
37 dmexg 7139 . . . . . . . . . 10 ((𝑔𝑓) ∈ V → dom (𝑔𝑓) ∈ V)
38 uniexg 6997 . . . . . . . . . 10 (dom (𝑔𝑓) ∈ V → dom (𝑔𝑓) ∈ V)
3936, 37, 383syl 18 . . . . . . . . 9 ((𝜑𝑓𝑋) → dom (𝑔𝑓) ∈ V)
4039ralrimivw 2996 . . . . . . . 8 ((𝜑𝑓𝑋) → ∀𝑔𝑋 dom (𝑔𝑓) ∈ V)
41 eqid 2651 . . . . . . . . 9 (𝑔𝑋 dom (𝑔𝑓)) = (𝑔𝑋 dom (𝑔𝑓))
4241fnmpt 6058 . . . . . . . 8 (∀𝑔𝑋 dom (𝑔𝑓) ∈ V → (𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋)
4340, 42syl 17 . . . . . . 7 ((𝜑𝑓𝑋) → (𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋)
44 dffn4 6159 . . . . . . 7 ((𝑔𝑋 dom (𝑔𝑓)) Fn 𝑋 ↔ (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓)))
4543, 44sylib 208 . . . . . 6 ((𝜑𝑓𝑋) → (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓)))
46 fonum 8919 . . . . . 6 ((𝑋 ∈ dom card ∧ (𝑔𝑋 dom (𝑔𝑓)):𝑋onto→ran (𝑔𝑋 dom (𝑔𝑓))) → ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card)
4724, 45, 46syl2anc 694 . . . . 5 ((𝜑𝑓𝑋) → ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card)
48 ssdif0 3975 . . . . . . . . . . . 12 ( (𝐹𝑘) ⊆ {(𝑓𝑘)} ↔ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅)
49 simpr 476 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) ⊆ {(𝑓𝑘)})
50 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓𝑋) → 𝑓𝑋)
5150, 20syl6eleq 2740 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓𝑋) → 𝑓X𝑘𝐴 (𝐹𝑘))
52 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑓 ∈ V
5352elixp 7957 . . . . . . . . . . . . . . . . . . . 20 (𝑓X𝑘𝐴 (𝐹𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘)))
5453simprbi 479 . . . . . . . . . . . . . . . . . . 19 (𝑓X𝑘𝐴 (𝐹𝑘) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
5551, 54syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓𝑋) → ∀𝑘𝐴 (𝑓𝑘) ∈ (𝐹𝑘))
5655r19.21bi 2961 . . . . . . . . . . . . . . . . 17 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (𝑓𝑘) ∈ (𝐹𝑘))
5756snssd 4372 . . . . . . . . . . . . . . . 16 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → {(𝑓𝑘)} ⊆ (𝐹𝑘))
5857adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → {(𝑓𝑘)} ⊆ (𝐹𝑘))
5949, 58eqssd 3653 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) = {(𝑓𝑘)})
60 fvex 6239 . . . . . . . . . . . . . . 15 (𝑓𝑘) ∈ V
6160ensn1 8061 . . . . . . . . . . . . . 14 {(𝑓𝑘)} ≈ 1𝑜
6259, 61syl6eqbr 4724 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ (𝐹𝑘) ⊆ {(𝑓𝑘)}) → (𝐹𝑘) ≈ 1𝑜)
6362ex 449 . . . . . . . . . . . 12 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → ( (𝐹𝑘) ⊆ {(𝑓𝑘)} → (𝐹𝑘) ≈ 1𝑜))
6448, 63syl5bir 233 . . . . . . . . . . 11 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅ → (𝐹𝑘) ≈ 1𝑜))
6564con3d 148 . . . . . . . . . 10 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1𝑜 → ¬ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅))
66 neq0 3963 . . . . . . . . . 10 (¬ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) = ∅ ↔ ∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}))
6765, 66syl6ib 241 . . . . . . . . 9 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1𝑜 → ∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})))
68 eldifi 3765 . . . . . . . . . . . . 13 (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → 𝑥 (𝐹𝑘))
69 simplr 807 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → 𝑥 (𝐹𝑘))
70 iftrue 4125 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = 𝑥)
7170, 18eleq12d 2724 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛) ↔ 𝑥 (𝐹𝑘)))
7269, 71syl5ibrcom 237 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
7350, 16syl6eleq 2740 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓𝑋) → 𝑓X𝑛𝐴 (𝐹𝑛))
7452elixp 7957 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓X𝑛𝐴 (𝐹𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛)))
7574simprbi 479 . . . . . . . . . . . . . . . . . . . . 21 (𝑓X𝑛𝐴 (𝐹𝑛) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7673, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓𝑋) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7776ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ∀𝑛𝐴 (𝑓𝑛) ∈ (𝐹𝑛))
7877r19.21bi 2961 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (𝑓𝑛) ∈ (𝐹𝑛))
79 iffalse 4128 . . . . . . . . . . . . . . . . . . 19 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = (𝑓𝑛))
8079eleq1d 2715 . . . . . . . . . . . . . . . . . 18 𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛) ↔ (𝑓𝑛) ∈ (𝐹𝑛)))
8178, 80syl5ibrcom 237 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛)))
8272, 81pm2.61d 170 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) ∧ 𝑛𝐴) → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛))
8382ralrimiva 2995 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → ∀𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ (𝐹𝑛))
84 ptcmp.3 . . . . . . . . . . . . . . . . 17 (𝜑𝐴𝑉)
8584ad3antrrr 766 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → 𝐴𝑉)
86 mptelixpg 7987 . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ X𝑛𝐴 (𝐹𝑛))
8988, 16syl6eleqr 2741 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 (𝐹𝑘)) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋)
9068, 89sylan2 490 . . . . . . . . . . . 12 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋)
91 vex 3234 . . . . . . . . . . . . . 14 𝑘 ∈ V
9291unisn 4483 . . . . . . . . . . . . 13 {𝑘} = 𝑘
93 simplr 807 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑘𝐴)
94 eleq1 2718 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑘 → (𝑚𝐴𝑘𝐴))
9593, 94syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘𝑚𝐴))
9695pm4.71rd 668 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚𝐴𝑚 = 𝑘)))
97 equequ1 1998 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑛 = 𝑘𝑚 = 𝑘))
98 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
9997, 98ifbieq2d 4144 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑚 → if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) = if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)))
100 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))
101 vex 3234 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥 ∈ V
102 fvex 6239 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓𝑚) ∈ V
103101, 102ifex 4189 . . . . . . . . . . . . . . . . . . . . . . 23 if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ∈ V
10499, 100, 103fvmpt 6321 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚𝐴 → ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) = if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)))
105104neeq1d 2882 . . . . . . . . . . . . . . . . . . . . 21 (𝑚𝐴 → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
106105adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
107 iffalse 4128 . . . . . . . . . . . . . . . . . . . . . 22 𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) = (𝑓𝑚))
108107necon1ai 2850 . . . . . . . . . . . . . . . . . . . . 21 (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) → 𝑚 = 𝑘)
109 eldifsni 4353 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → 𝑥 ≠ (𝑓𝑘))
110109ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → 𝑥 ≠ (𝑓𝑘))
111 iftrue 4125 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) = 𝑥)
112 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝑓𝑚) = (𝑓𝑘))
113111, 112neeq12d 2884 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑘 → (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) ↔ 𝑥 ≠ (𝑓𝑘)))
114110, 113syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚)))
115108, 114impbid2 216 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (if(𝑚 = 𝑘, 𝑥, (𝑓𝑚)) ≠ (𝑓𝑚) ↔ 𝑚 = 𝑘))
116106, 115bitrd 268 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) ∧ 𝑚𝐴) → (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚) ↔ 𝑚 = 𝑘))
117116pm5.32da 674 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → ((𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)) ↔ (𝑚𝐴𝑚 = 𝑘)))
11896, 117bitr4d 271 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))))
119118abbidv 2770 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑚𝑚 = 𝑘} = {𝑚 ∣ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))})
120 df-sn 4211 . . . . . . . . . . . . . . . 16 {𝑘} = {𝑚𝑚 = 𝑘}
121 df-rab 2950 . . . . . . . . . . . . . . . 16 {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)} = {𝑚 ∣ (𝑚𝐴 ∧ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚))}
122119, 120, 1213eqtr4g 2710 . . . . . . . . . . . . . . 15 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = {𝑚𝐴 ∣ ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)))‘𝑚) ≠ (𝑓𝑚)})
123 fvex 6239 . . . . . . . . . . . . . . . . . . 19 (𝑓𝑛) ∈ V
124101, 123ifex 4189 . . . . . . . . . . . . . . . . . 18 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V
125124rgenw 2953 . . . . . . . . . . . . . . . . 17 𝑛𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓𝑛)) ∈ V
126100fnmpt 6058 . . . . . . . . . . . . . . . . 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 7956 . . . . . . . . . . . . . . . . . 18 (𝑓X𝑛𝐴 (𝐹𝑛) → 𝑓 Fn 𝐴)
12973, 128syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓𝑋) → 𝑓 Fn 𝐴)
130129ad2antrr 762 . . . . . . . . . . . . . . . 16 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑓 Fn 𝐴)
131 fndmdif 6361 . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 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 2688 . . . . . . . . . . . . . 14 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
134133unieqd 4478 . . . . . . . . . . . . 13 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → {𝑘} = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
13592, 134syl5eqr 2699 . . . . . . . . . . . 12 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → 𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
136 difeq1 3754 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → (𝑔𝑓) = ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
137136dmeqd 5358 . . . . . . . . . . . . . . 15 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → dom (𝑔𝑓) = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
138137unieqd 4478 . . . . . . . . . . . . . 14 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → dom (𝑔𝑓) = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓))
139138eqeq2d 2661 . . . . . . . . . . . . 13 (𝑔 = (𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) → (𝑘 = dom (𝑔𝑓) ↔ 𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓)))
140139rspcev 3340 . . . . . . . . . . . 12 (((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∈ 𝑋𝑘 = dom ((𝑛𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓𝑛))) ∖ 𝑓)) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
14190, 135, 140syl2anc 694 . . . . . . . . . . 11 ((((𝜑𝑓𝑋) ∧ 𝑘𝐴) ∧ 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)})) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓))
142141ex 449 . . . . . . . . . 10 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
143142exlimdv 1901 . . . . . . . . 9 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (∃𝑥 𝑥 ∈ ( (𝐹𝑘) ∖ {(𝑓𝑘)}) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
14467, 143syld 47 . . . . . . . 8 (((𝜑𝑓𝑋) ∧ 𝑘𝐴) → (¬ (𝐹𝑘) ≈ 1𝑜 → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
145144expimpd 628 . . . . . . 7 ((𝜑𝑓𝑋) → ((𝑘𝐴 ∧ ¬ (𝐹𝑘) ≈ 1𝑜) → ∃𝑔𝑋 𝑘 = dom (𝑔𝑓)))
14618breq1d 4695 . . . . . . . . 9 (𝑛 = 𝑘 → ( (𝐹𝑛) ≈ 1𝑜 (𝐹𝑘) ≈ 1𝑜))
147146notbid 307 . . . . . . . 8 (𝑛 = 𝑘 → (¬ (𝐹𝑛) ≈ 1𝑜 ↔ ¬ (𝐹𝑘) ≈ 1𝑜))
148147elrab 3396 . . . . . . 7 (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ↔ (𝑘𝐴 ∧ ¬ (𝐹𝑘) ≈ 1𝑜))
14941elrnmpt 5404 . . . . . . . 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 285 . . . . . 6 ((𝜑𝑓𝑋) → (𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} → 𝑘 ∈ ran (𝑔𝑋 dom (𝑔𝑓))))
152151ssrdv 3642 . . . . 5 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ ran (𝑔𝑋 dom (𝑔𝑓)))
153 ssnum 8900 . . . . 5 ((ran (𝑔𝑋 dom (𝑔𝑓)) ∈ dom card ∧ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ ran (𝑔𝑋 dom (𝑔𝑓))) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ dom card)
15447, 152, 153syl2anc 694 . . . 4 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ dom card)
155 xpnum 8815 . . . 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 694 . . 3 ((𝜑𝑓𝑋) → (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ dom card)
15784adantr 480 . . . . 5 ((𝜑𝑓𝑋) → 𝐴𝑉)
158 rabexg 4844 . . . . 5 (𝐴𝑉 → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ V)
159157, 158syl 17 . . . 4 ((𝜑𝑓𝑋) → {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ V)
160 fvex 6239 . . . . . . 7 (𝐹𝑘) ∈ V
161160uniex 6995 . . . . . 6 (𝐹𝑘) ∈ V
162161rgenw 2953 . . . . 5 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V
163 iunexg 7185 . . . . 5 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ∈ V ∧ ∀𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V)
164159, 162, 163sylancl 695 . . . 4 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ V)
165 resixp 7985 . . . . . 6 (({𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} ⊆ 𝐴𝑓X𝑘𝐴 (𝐹𝑘)) → (𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
16626, 51, 165sylancr 696 . . . . 5 ((𝜑𝑓𝑋) → (𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘))
167 ne0i 3954 . . . . 5 ((𝑓 ↾ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}) ∈ X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ≠ ∅)
168166, 167syl 17 . . . 4 ((𝜑𝑓𝑋) → X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ≠ ∅)
169 ixpiunwdom 8537 . . . 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}))
170159, 164, 168, 169syl3anc 1366 . . 3 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ≼* (X𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) × {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜}))
171 numwdom 8920 . . 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)
172156, 170, 171syl2anc 694 . 2 ((𝜑𝑓𝑋) → 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
17315, 172exlimddv 1903 1 (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  cmpt2 6692  1𝑜c1o 7598  Xcixp 7950  cen 7994  Fincfn 7997  * cwdom 8503  cardccrd 8799  Compccmp 21237  UFLcufl 21751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-fin 8001  df-wdom 8505  df-card 8803  df-acn 8806
This theorem is referenced by:  ptcmplem3  21905
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