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Theorem cantnflem3 9729
Description: Lemma for cantnf 9731. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 8640 to factor 𝐶 into the form ((𝐴o 𝑋) ·o 𝑌) +o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴o 𝑋) ≤ (𝐴o 𝑋) ·o 𝑌𝐶, 𝑍 has a normal form, and by appending the term (𝐴o 𝑋) ·o 𝑌 using cantnfp1 9719 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
cantnf.c (𝜑𝐶 ∈ (𝐴o 𝐵))
cantnf.s (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e (𝜑 → ∅ ∈ 𝐶)
cantnf.x 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴o 𝑐)}
cantnf.p 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
cantnf.y 𝑌 = (1st𝑃)
cantnf.z 𝑍 = (2nd𝑃)
cantnf.g (𝜑𝐺𝑆)
cantnf.v (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
cantnf.f 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
Assertion
Ref Expression
cantnflem3 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Distinct variable groups:   𝑡,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧,𝐶   𝑡,𝑎,𝐴,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑡   𝑤,𝐹,𝑥,𝑦,𝑧   𝑆,𝑐,𝑡,𝑥,𝑦,𝑧   𝑡,𝑍,𝑥,𝑦,𝑧   𝐺,𝑐,𝑡,𝑤,𝑥,𝑦,𝑧   𝜑,𝑡,𝑥,𝑦,𝑧   𝑡,𝑌,𝑤,𝑥,𝑦,𝑧   𝑋,𝑎,𝑏,𝑑,𝑡,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑑)   𝐶(𝑡)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑡,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑤,𝑎,𝑏,𝑑)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑑)   𝐹(𝑡,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑎,𝑏,𝑑)   𝑋(𝑐)   𝑌(𝑎,𝑏,𝑐,𝑑)   𝑍(𝑤,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem cantnflem3
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 cantnfs.s . . . . 5 𝑆 = dom (𝐴 CNF 𝐵)
2 cantnfs.a . . . . 5 (𝜑𝐴 ∈ On)
3 cantnfs.b . . . . 5 (𝜑𝐵 ∈ On)
4 cantnf.g . . . . 5 (𝜑𝐺𝑆)
5 oemapval.t . . . . . . . . . . . . . 14 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
6 cantnf.c . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ (𝐴o 𝐵))
7 cantnf.s . . . . . . . . . . . . . 14 (𝜑𝐶 ⊆ ran (𝐴 CNF 𝐵))
8 cantnf.e . . . . . . . . . . . . . 14 (𝜑 → ∅ ∈ 𝐶)
91, 2, 3, 5, 6, 7, 8cantnflem2 9728 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
10 eqid 2735 . . . . . . . . . . . . . . 15 𝑋 = 𝑋
11 eqid 2735 . . . . . . . . . . . . . . 15 𝑌 = 𝑌
12 eqid 2735 . . . . . . . . . . . . . . 15 𝑍 = 𝑍
1310, 11, 123pm3.2i 1338 . . . . . . . . . . . . . 14 (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)
14 cantnf.x . . . . . . . . . . . . . . 15 𝑋 = {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴o 𝑐)}
15 cantnf.p . . . . . . . . . . . . . . 15 𝑃 = (℩𝑑𝑎 ∈ On ∃𝑏 ∈ (𝐴o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
16 cantnf.y . . . . . . . . . . . . . . 15 𝑌 = (1st𝑃)
17 cantnf.z . . . . . . . . . . . . . . 15 𝑍 = (2nd𝑃)
1814, 15, 16, 17oeeui 8639 . . . . . . . . . . . . . 14 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)) ∧ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶) ↔ (𝑋 = 𝑋𝑌 = 𝑌𝑍 = 𝑍)))
1913, 18mpbiri 258 . . . . . . . . . . . . 13 ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)) ∧ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
209, 19syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)) ∧ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
2120simpld 494 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)))
2221simp1d 1141 . . . . . . . . . 10 (𝜑𝑋 ∈ On)
23 oecl 8574 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
242, 22, 23syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐴o 𝑋) ∈ On)
2521simp2d 1142 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐴 ∖ 1o))
2625eldifad 3975 . . . . . . . . . 10 (𝜑𝑌𝐴)
27 onelon 6411 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
282, 26, 27syl2anc 584 . . . . . . . . 9 (𝜑𝑌 ∈ On)
29 dif1o 8537 . . . . . . . . . . . 12 (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌𝐴𝑌 ≠ ∅))
3029simprbi 496 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
3125, 30syl 17 . . . . . . . . . 10 (𝜑𝑌 ≠ ∅)
32 on0eln0 6442 . . . . . . . . . . 11 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3328, 32syl 17 . . . . . . . . . 10 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
3431, 33mpbird 257 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝑌)
35 omword1 8610 . . . . . . . . 9 ((((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
3624, 28, 34, 35syl21anc 838 . . . . . . . 8 (𝜑 → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
37 omcl 8573 . . . . . . . . . . 11 (((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3824, 28, 37syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3921simp3d 1143 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝐴o 𝑋))
40 onelon 6411 . . . . . . . . . . 11 (((𝐴o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴o 𝑋)) → 𝑍 ∈ On)
4124, 39, 40syl2anc 584 . . . . . . . . . 10 (𝜑𝑍 ∈ On)
42 oaword1 8589 . . . . . . . . . 10 ((((𝐴o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
4338, 41, 42syl2anc 584 . . . . . . . . 9 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
4420simprd 495 . . . . . . . . 9 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
4543, 44sseqtrd 4036 . . . . . . . 8 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ 𝐶)
4636, 45sstrd 4006 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ⊆ 𝐶)
47 oecl 8574 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
482, 3, 47syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴o 𝐵) ∈ On)
49 ontr2 6433 . . . . . . . 8 (((𝐴o 𝑋) ∈ On ∧ (𝐴o 𝐵) ∈ On) → (((𝐴o 𝑋) ⊆ 𝐶𝐶 ∈ (𝐴o 𝐵)) → (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5024, 48, 49syl2anc 584 . . . . . . 7 (𝜑 → (((𝐴o 𝑋) ⊆ 𝐶𝐶 ∈ (𝐴o 𝐵)) → (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5146, 6, 50mp2and 699 . . . . . 6 (𝜑 → (𝐴o 𝑋) ∈ (𝐴o 𝐵))
529simpld 494 . . . . . . 7 (𝜑𝐴 ∈ (On ∖ 2o))
53 oeord 8625 . . . . . . 7 ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑋𝐵 ↔ (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5422, 3, 52, 53syl3anc 1370 . . . . . 6 (𝜑 → (𝑋𝐵 ↔ (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5551, 54mpbird 257 . . . . 5 (𝜑𝑋𝐵)
562adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ On)
573adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐵 ∈ On)
58 suppssdm 8201 . . . . . . . . . . . . . . 15 (𝐺 supp ∅) ⊆ dom 𝐺
591, 2, 3cantnfs 9704 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
604, 59mpbid 232 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6160simpld 494 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝐵𝐴)
6258, 61fssdm 6756 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6362sselda 3995 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝐵)
64 onelon 6411 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6557, 63, 64syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
66 oecl 8574 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴o 𝑥) ∈ On)
6756, 65, 66syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ∈ On)
6861adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵𝐴)
6968, 63ffvelcdmd 7105 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ 𝐴)
70 onelon 6411 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐺𝑥) ∈ 𝐴) → (𝐺𝑥) ∈ On)
7156, 69, 70syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ On)
7261ffnd 6738 . . . . . . . . . . . . . 14 (𝜑𝐺 Fn 𝐵)
738elexd 3502 . . . . . . . . . . . . . 14 (𝜑 → ∅ ∈ V)
74 elsuppfn 8194 . . . . . . . . . . . . . 14 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7572, 3, 73, 74syl3anc 1370 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7675simplbda 499 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ≠ ∅)
77 on0eln0 6442 . . . . . . . . . . . . 13 ((𝐺𝑥) ∈ On → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
7871, 77syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
7976, 78mpbird 257 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺𝑥))
80 omword1 8610 . . . . . . . . . . 11 ((((𝐴o 𝑥) ∈ On ∧ (𝐺𝑥) ∈ On) ∧ ∅ ∈ (𝐺𝑥)) → (𝐴o 𝑥) ⊆ ((𝐴o 𝑥) ·o (𝐺𝑥)))
8167, 71, 79, 80syl21anc 838 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ⊆ ((𝐴o 𝑥) ·o (𝐺𝑥)))
82 eqid 2735 . . . . . . . . . . . 12 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
834adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺𝑆)
84 eqid 2735 . . . . . . . . . . . 12 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·o (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·o (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +o 𝑧)), ∅)
851, 56, 57, 82, 83, 84, 63cantnfle 9709 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴o 𝑥) ·o (𝐺𝑥)) ⊆ ((𝐴 CNF 𝐵)‘𝐺))
86 cantnf.v . . . . . . . . . . . 12 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
8786adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
8885, 87sseqtrd 4036 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴o 𝑥) ·o (𝐺𝑥)) ⊆ 𝑍)
8981, 88sstrd 4006 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ⊆ 𝑍)
9039adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴o 𝑋))
9124adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑋) ∈ On)
92 ontr2 6433 . . . . . . . . . 10 (((𝐴o 𝑥) ∈ On ∧ (𝐴o 𝑋) ∈ On) → (((𝐴o 𝑥) ⊆ 𝑍𝑍 ∈ (𝐴o 𝑋)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9367, 91, 92syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (((𝐴o 𝑥) ⊆ 𝑍𝑍 ∈ (𝐴o 𝑋)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9489, 90, 93mp2and 699 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋))
9522adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
9652adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2o))
97 oeord 8625 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑥𝑋 ↔ (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9865, 95, 96, 97syl3anc 1370 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝑥𝑋 ↔ (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9994, 98mpbird 257 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝑋)
10099ex 412 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥𝑋))
101100ssrdv 4001 . . . . 5 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
102 cantnf.f . . . . 5 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
1031, 2, 3, 4, 55, 26, 101, 102cantnfp1 9719 . . . 4 (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
104103simprd 495 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
10586oveq2d 7447 . . 3 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
106104, 105, 443eqtrd 2779 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
1071, 2, 3cantnff 9712 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
108107ffnd 6738 . . 3 (𝜑 → (𝐴 CNF 𝐵) Fn 𝑆)
109103simpld 494 . . 3 (𝜑𝐹𝑆)
110 fnfvelrn 7100 . . 3 (((𝐴 CNF 𝐵) Fn 𝑆𝐹𝑆) → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
111108, 109, 110syl2anc 584 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
112106, 111eqeltrrd 2840 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  c0 4339  ifcif 4531  cop 4637   cuni 4912   cint 4951   class class class wbr 5148  {copab 5210  cmpt 5231   E cep 5588  dom cdm 5689  ran crn 5690  Oncon0 6386  cio 6514   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012   supp csupp 8184  seqωcseqom 8486  1oc1o 8498  2oc2o 8499   +o coa 8502   ·o comu 8503  o coe 8504   finSupp cfsupp 9399  OrdIsocoi 9547   CNF ccnf 9699
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-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700
This theorem is referenced by:  cantnflem4  9730
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