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Theorem wfrlem15 7414
Description: Lemma for well-founded recursion. When 𝑧 is 𝑅 minimal, 𝐶 is an acceptable function. This step is where the Axiom of Replacement becomes required. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem13.1 𝑅 We 𝐴
wfrlem13.2 𝑅 Se 𝐴
wfrlem13.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13.4 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
Assertion
Ref Expression
wfrlem15 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐹,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦,𝑧   𝐶,𝑓,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem15
StepHypRef Expression
1 wfrlem13.1 . . . . . 6 𝑅 We 𝐴
2 wfrlem13.3 . . . . . 6 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
31, 2wfrlem10 7409 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
4 eldifi 3724 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
5 wfrlem13.2 . . . . . . 7 𝑅 Se 𝐴
6 setlikespec 5689 . . . . . . 7 ((𝑧𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
74, 5, 6sylancl 693 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
87adantr 481 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
93, 8eqeltrrd 2700 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐹 ∈ V)
10 snex 4899 . . . 4 {𝑧} ∈ V
11 unexg 6944 . . . 4 ((dom 𝐹 ∈ V ∧ {𝑧} ∈ V) → (dom 𝐹 ∪ {𝑧}) ∈ V)
129, 10, 11sylancl 693 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ∈ V)
13 wfrlem13.4 . . . . . 6 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
141, 5, 2, 13wfrlem13 7412 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
1514adantr 481 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
162wfrdmss 7406 . . . . . . 7 dom 𝐹𝐴
174snssd 4331 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → {𝑧} ⊆ 𝐴)
18 unss 3779 . . . . . . . 8 ((dom 𝐹𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
1918biimpi 206 . . . . . . 7 ((dom 𝐹𝐴 ∧ {𝑧} ⊆ 𝐴) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
2016, 17, 19sylancr 694 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
2120adantr 481 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
22 elun 3745 . . . . . . . 8 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}))
23 velsn 4184 . . . . . . . . 9 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
2423orbi2i 541 . . . . . . . 8 ((𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
2522, 24bitri 264 . . . . . . 7 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
262wfrdmcl 7408 . . . . . . . . . 10 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
27 ssun3 3770 . . . . . . . . . 10 (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
2826, 27syl 17 . . . . . . . . 9 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
2928a1i 11 . . . . . . . 8 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
30 ssun1 3768 . . . . . . . . . 10 dom 𝐹 ⊆ (dom 𝐹 ∪ {𝑧})
313, 30syl6eqss 3647 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧}))
32 predeq3 5672 . . . . . . . . . 10 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
3332sseq1d 3624 . . . . . . . . 9 (𝑦 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧})))
3431, 33syl5ibrcom 237 . . . . . . . 8 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
3529, 34jaod 395 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ((𝑦 ∈ dom 𝐹𝑦 = 𝑧) → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
3625, 35syl5bi 232 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
3736ralrimiv 2962 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
3821, 37jca 554 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
391, 5, 2, 13wfrlem14 7413 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
4039ralrimiv 2962 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
4140adantr 481 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
4215, 38, 413jca 1240 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
43 fneq2 5968 . . . . 5 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝐶 Fn 𝑥𝐶 Fn (dom 𝐹 ∪ {𝑧})))
44 sseq1 3618 . . . . . 6 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝑥𝐴 ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴))
45 sseq2 3619 . . . . . . 7 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
4645raleqbi1dv 3141 . . . . . 6 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
4744, 46anbi12d 746 . . . . 5 (𝑥 = (dom 𝐹 ∪ {𝑧}) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))))
48 raleq 3133 . . . . 5 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
4943, 47, 483anbi123d 1397 . . . 4 (𝑥 = (dom 𝐹 ∪ {𝑧}) → ((𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5049spcegv 3289 . . 3 ((dom 𝐹 ∪ {𝑧}) ∈ V → ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5112, 42, 50sylc 65 . 2 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
5210, 11mpan2 706 . . . . 5 (dom 𝐹 ∈ V → (dom 𝐹 ∪ {𝑧}) ∈ V)
53 fnex 6466 . . . . 5 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ (dom 𝐹 ∪ {𝑧}) ∈ V) → 𝐶 ∈ V)
5452, 53sylan2 491 . . . 4 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ dom 𝐹 ∈ V) → 𝐶 ∈ V)
5515, 9, 54syl2anc 692 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ V)
56 fneq1 5967 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
57 fveq1 6177 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
58 reseq1 5379 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))
5958fveq2d 6182 . . . . . . . 8 (𝑓 = 𝐶 → (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
6057, 59eqeq12d 2635 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
6160ralbidv 2983 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
6256, 613anbi13d 1399 . . . . 5 (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
6362exbidv 1848 . . . 4 (𝑓 = 𝐶 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
6463elabg 3345 . . 3 (𝐶 ∈ V → (𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
6555, 64syl 17 . 2 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
6651, 65mpbird 247 1 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  Vcvv 3195  cdif 3564  cun 3565  wss 3567  c0 3907  {csn 4168  cop 4174   Se wse 5061   We wwe 5062  dom cdm 5104  cres 5106  Predcpred 5667   Fn wfn 5871  cfv 5876  wrecscwrecs 7391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-wrecs 7392
This theorem is referenced by:  wfrlem16  7415
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