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Theorem cantnflem3 9142
Description: Lemma for cantnf 9144. 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 8216 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 9132 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 9141 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
10 eqid 2801 . . . . . . . . . . . . . . 15 𝑋 = 𝑋
11 eqid 2801 . . . . . . . . . . . . . . 15 𝑌 = 𝑌
12 eqid 2801 . . . . . . . . . . . . . . 15 𝑍 = 𝑍
1310, 11, 123pm3.2i 1336 . . . . . . . . . . . . . 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 8215 . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . 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 498 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)))
2221simp1d 1139 . . . . . . . . . 10 (𝜑𝑋 ∈ On)
23 oecl 8149 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
242, 22, 23syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐴o 𝑋) ∈ On)
2521simp2d 1140 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐴 ∖ 1o))
2625eldifad 3896 . . . . . . . . . 10 (𝜑𝑌𝐴)
27 onelon 6188 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
282, 26, 27syl2anc 587 . . . . . . . . 9 (𝜑𝑌 ∈ On)
29 dif1o 8112 . . . . . . . . . . . 12 (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌𝐴𝑌 ≠ ∅))
3029simprbi 500 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
3125, 30syl 17 . . . . . . . . . 10 (𝜑𝑌 ≠ ∅)
32 on0eln0 6218 . . . . . . . . . . 11 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3328, 32syl 17 . . . . . . . . . 10 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
3431, 33mpbird 260 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝑌)
35 omword1 8186 . . . . . . . . 9 ((((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
3624, 28, 34, 35syl21anc 836 . . . . . . . 8 (𝜑 → (𝐴o 𝑋) ⊆ ((𝐴o 𝑋) ·o 𝑌))
37 omcl 8148 . . . . . . . . . . 11 (((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3824, 28, 37syl2anc 587 . . . . . . . . . 10 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3921simp3d 1141 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝐴o 𝑋))
40 onelon 6188 . . . . . . . . . . 11 (((𝐴o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴o 𝑋)) → 𝑍 ∈ On)
4124, 39, 40syl2anc 587 . . . . . . . . . 10 (𝜑𝑍 ∈ On)
42 oaword1 8165 . . . . . . . . . 10 ((((𝐴o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
4338, 41, 42syl2anc 587 . . . . . . . . 9 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
4420simprd 499 . . . . . . . . 9 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
4543, 44sseqtrd 3958 . . . . . . . 8 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ 𝐶)
4636, 45sstrd 3928 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ⊆ 𝐶)
47 oecl 8149 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
482, 3, 47syl2anc 587 . . . . . . . 8 (𝜑 → (𝐴o 𝐵) ∈ On)
49 ontr2 6210 . . . . . . . 8 (((𝐴o 𝑋) ∈ On ∧ (𝐴o 𝐵) ∈ On) → (((𝐴o 𝑋) ⊆ 𝐶𝐶 ∈ (𝐴o 𝐵)) → (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5024, 48, 49syl2anc 587 . . . . . . 7 (𝜑 → (((𝐴o 𝑋) ⊆ 𝐶𝐶 ∈ (𝐴o 𝐵)) → (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5146, 6, 50mp2and 698 . . . . . 6 (𝜑 → (𝐴o 𝑋) ∈ (𝐴o 𝐵))
529simpld 498 . . . . . . 7 (𝜑𝐴 ∈ (On ∖ 2o))
53 oeord 8201 . . . . . . 7 ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑋𝐵 ↔ (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5422, 3, 52, 53syl3anc 1368 . . . . . 6 (𝜑 → (𝑋𝐵 ↔ (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5551, 54mpbird 260 . . . . 5 (𝜑𝑋𝐵)
562adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ On)
573adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐵 ∈ On)
58 suppssdm 7830 . . . . . . . . . . . . . . 15 (𝐺 supp ∅) ⊆ dom 𝐺
591, 2, 3cantnfs 9117 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
604, 59mpbid 235 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6160simpld 498 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝐵𝐴)
6258, 61fssdm 6508 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6362sselda 3918 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝐵)
64 onelon 6188 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6557, 63, 64syl2anc 587 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
66 oecl 8149 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴o 𝑥) ∈ On)
6756, 65, 66syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ∈ On)
6861adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵𝐴)
6968, 63ffvelrnd 6833 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ 𝐴)
70 onelon 6188 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐺𝑥) ∈ 𝐴) → (𝐺𝑥) ∈ On)
7156, 69, 70syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ On)
7261ffnd 6492 . . . . . . . . . . . . . 14 (𝜑𝐺 Fn 𝐵)
738elexd 3464 . . . . . . . . . . . . . 14 (𝜑 → ∅ ∈ V)
74 elsuppfn 7825 . . . . . . . . . . . . . 14 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7572, 3, 73, 74syl3anc 1368 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7675simplbda 503 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ≠ ∅)
77 on0eln0 6218 . . . . . . . . . . . . 13 ((𝐺𝑥) ∈ On → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
7871, 77syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
7976, 78mpbird 260 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺𝑥))
80 omword1 8186 . . . . . . . . . . 11 ((((𝐴o 𝑥) ∈ On ∧ (𝐺𝑥) ∈ On) ∧ ∅ ∈ (𝐺𝑥)) → (𝐴o 𝑥) ⊆ ((𝐴o 𝑥) ·o (𝐺𝑥)))
8167, 71, 79, 80syl21anc 836 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ⊆ ((𝐴o 𝑥) ·o (𝐺𝑥)))
82 eqid 2801 . . . . . . . . . . . 12 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
834adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺𝑆)
84 eqid 2801 . . . . . . . . . . . 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 9122 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴o 𝑥) ·o (𝐺𝑥)) ⊆ ((𝐴 CNF 𝐵)‘𝐺))
86 cantnf.v . . . . . . . . . . . 12 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
8786adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
8885, 87sseqtrd 3958 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴o 𝑥) ·o (𝐺𝑥)) ⊆ 𝑍)
8981, 88sstrd 3928 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ⊆ 𝑍)
9039adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴o 𝑋))
9124adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑋) ∈ On)
92 ontr2 6210 . . . . . . . . . 10 (((𝐴o 𝑥) ∈ On ∧ (𝐴o 𝑋) ∈ On) → (((𝐴o 𝑥) ⊆ 𝑍𝑍 ∈ (𝐴o 𝑋)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9367, 91, 92syl2anc 587 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (((𝐴o 𝑥) ⊆ 𝑍𝑍 ∈ (𝐴o 𝑋)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9489, 90, 93mp2and 698 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋))
9522adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
9652adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2o))
97 oeord 8201 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑥𝑋 ↔ (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9865, 95, 96, 97syl3anc 1368 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝑥𝑋 ↔ (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9994, 98mpbird 260 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝑋)
10099ex 416 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥𝑋))
101100ssrdv 3924 . . . . 5 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
102 cantnf.f . . . . 5 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
1031, 2, 3, 4, 55, 26, 101, 102cantnfp1 9132 . . . 4 (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
104103simprd 499 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
10586oveq2d 7155 . . 3 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
106104, 105, 443eqtrd 2840 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
1071, 2, 3cantnff 9125 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
108107ffnd 6492 . . 3 (𝜑 → (𝐴 CNF 𝐵) Fn 𝑆)
109103simpld 498 . . 3 (𝜑𝐹𝑆)
110 fnfvelrn 6829 . . 3 (((𝐴 CNF 𝐵) Fn 𝑆𝐹𝑆) → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
111108, 109, 110syl2anc 587 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
112106, 111eqeltrrd 2894 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cdif 3881  wss 3884  c0 4246  ifcif 4428  cop 4534   cuni 4803   cint 4841   class class class wbr 5033  {copab 5095  cmpt 5113   E cep 5432  dom cdm 5523  ran crn 5524  Oncon0 6163  cio 6285   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673  2nd c2nd 7674   supp csupp 7817  seqωcseqom 8070  1oc1o 8082  2oc2o 8083   +o coa 8086   ·o comu 8087  o coe 8088   finSupp cfsupp 8821  OrdIsocoi 8961   CNF ccnf 9112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-cnf 9113
This theorem is referenced by:  cantnflem4  9143
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