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Theorem wfrlem15 7581
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 7576 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
4 eldifi 3881 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
5 wfrlem13.2 . . . . . . 7 𝑅 Se 𝐴
6 setlikespec 5844 . . . . . . 7 ((𝑧𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
74, 5, 6sylancl 566 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
87adantr 466 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
93, 8eqeltrrd 2850 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐹 ∈ V)
10 snex 5036 . . . 4 {𝑧} ∈ V
11 unexg 7105 . . . 4 ((dom 𝐹 ∈ V ∧ {𝑧} ∈ V) → (dom 𝐹 ∪ {𝑧}) ∈ V)
129, 10, 11sylancl 566 . . 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 7579 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
1514adantr 466 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
162wfrdmss 7573 . . . . . . 7 dom 𝐹𝐴
174snssd 4473 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → {𝑧} ⊆ 𝐴)
18 unss 3936 . . . . . . . 8 ((dom 𝐹𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
1918biimpi 206 . . . . . . 7 ((dom 𝐹𝐴 ∧ {𝑧} ⊆ 𝐴) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
2016, 17, 19sylancr 567 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
2120adantr 466 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
22 elun 3902 . . . . . . . 8 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}))
23 velsn 4330 . . . . . . . . 9 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
2423orbi2i 877 . . . . . . . 8 ((𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
2522, 24bitri 264 . . . . . . 7 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
262wfrdmcl 7575 . . . . . . . . . 10 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
27 ssun3 3927 . . . . . . . . . 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 3925 . . . . . . . . . 10 dom 𝐹 ⊆ (dom 𝐹 ∪ {𝑧})
313, 30syl6eqss 3802 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧}))
32 predeq3 5827 . . . . . . . . . 10 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
3332sseq1d 3779 . . . . . . . . 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 839 . . . . . . 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 3113 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
3821, 37jca 495 . . . 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 7580 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
4039ralrimiv 3113 . . . . 5 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
4140adantr 466 . . . 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 1121 . . 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 6120 . . . . 5 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝐶 Fn 𝑥𝐶 Fn (dom 𝐹 ∪ {𝑧})))
44 sseq1 3773 . . . . . 6 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝑥𝐴 ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴))
45 sseq2 3774 . . . . . . 7 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
4645raleqbi1dv 3294 . . . . . 6 (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
4744, 46anbi12d 608 . . . . 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 3286 . . . . 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 1546 . . . 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 3443 . . 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 663 . . . . 5 (dom 𝐹 ∈ V → (dom 𝐹 ∪ {𝑧}) ∈ V)
53 fnex 6624 . . . . 5 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ (dom 𝐹 ∪ {𝑧}) ∈ V) → 𝐶 ∈ V)
5452, 53sylan2 572 . . . 4 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ dom 𝐹 ∈ V) → 𝐶 ∈ V)
5515, 9, 54syl2anc 565 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ V)
56 fneq1 6119 . . . . . 6 (𝑓 = 𝐶 → (𝑓 Fn 𝑥𝐶 Fn 𝑥))
57 fveq1 6331 . . . . . . . 8 (𝑓 = 𝐶 → (𝑓𝑦) = (𝐶𝑦))
58 reseq1 5528 . . . . . . . . 9 (𝑓 = 𝐶 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))
5958fveq2d 6336 . . . . . . . 8 (𝑓 = 𝐶 → (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
6057, 59eqeq12d 2785 . . . . . . 7 (𝑓 = 𝐶 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
6160ralbidv 3134 . . . . . 6 (𝑓 = 𝐶 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
6256, 613anbi13d 1548 . . . . 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 2001 . . . 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 3500 . . 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  wa 382  wo 826  w3a 1070   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wral 3060  Vcvv 3349  cdif 3718  cun 3719  wss 3721  c0 4061  {csn 4314  cop 4320   Se wse 5206   We wwe 5207  dom cdm 5249  cres 5251  Predcpred 5822   Fn wfn 6026  cfv 6031  wrecscwrecs 7557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7558
This theorem is referenced by:  wfrlem16  7582
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