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Theorem cantnflem3 9627
Description: Lemma for cantnf 9629. 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 8550 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 9617 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 9626 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
10 eqid 2736 . . . . . . . . . . . . . . 15 𝑋 = 𝑋
11 eqid 2736 . . . . . . . . . . . . . . 15 𝑌 = 𝑌
12 eqid 2736 . . . . . . . . . . . . . . 15 𝑍 = 𝑍
1310, 11, 123pm3.2i 1339 . . . . . . . . . . . . . 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 8549 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 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 495 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴o 𝑋)))
2221simp1d 1142 . . . . . . . . . 10 (𝜑𝑋 ∈ On)
23 oecl 8483 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
242, 22, 23syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐴o 𝑋) ∈ On)
2521simp2d 1143 . . . . . . . . . . 11 (𝜑𝑌 ∈ (𝐴 ∖ 1o))
2625eldifad 3922 . . . . . . . . . 10 (𝜑𝑌𝐴)
27 onelon 6342 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
282, 26, 27syl2anc 584 . . . . . . . . 9 (𝜑𝑌 ∈ On)
29 dif1o 8446 . . . . . . . . . . . 12 (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌𝐴𝑌 ≠ ∅))
3029simprbi 497 . . . . . . . . . . 11 (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
3125, 30syl 17 . . . . . . . . . 10 (𝜑𝑌 ≠ ∅)
32 on0eln0 6373 . . . . . . . . . . 11 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
3328, 32syl 17 . . . . . . . . . 10 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
3431, 33mpbird 256 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝑌)
35 omword1 8520 . . . . . . . . 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 8482 . . . . . . . . . . 11 (((𝐴o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3824, 28, 37syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ∈ On)
3921simp3d 1144 . . . . . . . . . . 11 (𝜑𝑍 ∈ (𝐴o 𝑋))
40 onelon 6342 . . . . . . . . . . 11 (((𝐴o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴o 𝑋)) → 𝑍 ∈ On)
4124, 39, 40syl2anc 584 . . . . . . . . . 10 (𝜑𝑍 ∈ On)
42 oaword1 8499 . . . . . . . . . 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 496 . . . . . . . . 9 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
4543, 44sseqtrd 3984 . . . . . . . 8 (𝜑 → ((𝐴o 𝑋) ·o 𝑌) ⊆ 𝐶)
4636, 45sstrd 3954 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ⊆ 𝐶)
47 oecl 8483 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
482, 3, 47syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴o 𝐵) ∈ On)
49 ontr2 6364 . . . . . . . 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 697 . . . . . 6 (𝜑 → (𝐴o 𝑋) ∈ (𝐴o 𝐵))
529simpld 495 . . . . . . 7 (𝜑𝐴 ∈ (On ∖ 2o))
53 oeord 8535 . . . . . . 7 ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑋𝐵 ↔ (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5422, 3, 52, 53syl3anc 1371 . . . . . 6 (𝜑 → (𝑋𝐵 ↔ (𝐴o 𝑋) ∈ (𝐴o 𝐵)))
5551, 54mpbird 256 . . . . 5 (𝜑𝑋𝐵)
562adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ On)
573adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐵 ∈ On)
58 suppssdm 8108 . . . . . . . . . . . . . . 15 (𝐺 supp ∅) ⊆ dom 𝐺
591, 2, 3cantnfs 9602 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
604, 59mpbid 231 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6160simpld 495 . . . . . . . . . . . . . . 15 (𝜑𝐺:𝐵𝐴)
6258, 61fssdm 6688 . . . . . . . . . . . . . 14 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6362sselda 3944 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝐵)
64 onelon 6342 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
6557, 63, 64syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
66 oecl 8483 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴o 𝑥) ∈ On)
6756, 65, 66syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ∈ On)
6861adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵𝐴)
6968, 63ffvelcdmd 7036 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ 𝐴)
70 onelon 6342 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝐺𝑥) ∈ 𝐴) → (𝐺𝑥) ∈ On)
7156, 69, 70syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ∈ On)
7261ffnd 6669 . . . . . . . . . . . . . 14 (𝜑𝐺 Fn 𝐵)
738elexd 3465 . . . . . . . . . . . . . 14 (𝜑 → ∅ ∈ V)
74 elsuppfn 8102 . . . . . . . . . . . . . 14 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7572, 3, 73, 74syl3anc 1371 . . . . . . . . . . . . 13 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥𝐵 ∧ (𝐺𝑥) ≠ ∅)))
7675simplbda 500 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐺𝑥) ≠ ∅)
77 on0eln0 6373 . . . . . . . . . . . . 13 ((𝐺𝑥) ∈ On → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
7871, 77syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (∅ ∈ (𝐺𝑥) ↔ (𝐺𝑥) ≠ ∅))
7976, 78mpbird 256 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺𝑥))
80 omword1 8520 . . . . . . . . . . 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 2736 . . . . . . . . . . . 12 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
834adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐺𝑆)
84 eqid 2736 . . . . . . . . . . . 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 9607 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴o 𝑥) ·o (𝐺𝑥)) ⊆ ((𝐴 CNF 𝐵)‘𝐺))
86 cantnf.v . . . . . . . . . . . 12 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
8786adantr 481 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
8885, 87sseqtrd 3984 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → ((𝐴o 𝑥) ·o (𝐺𝑥)) ⊆ 𝑍)
8981, 88sstrd 3954 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ⊆ 𝑍)
9039adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴o 𝑋))
9124adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑋) ∈ On)
92 ontr2 6364 . . . . . . . . . 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 697 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝐴o 𝑥) ∈ (𝐴o 𝑋))
9522adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
9652adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2o))
97 oeord 8535 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑥𝑋 ↔ (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9865, 95, 96, 97syl3anc 1371 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → (𝑥𝑋 ↔ (𝐴o 𝑥) ∈ (𝐴o 𝑋)))
9994, 98mpbird 256 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 supp ∅)) → 𝑥𝑋)
10099ex 413 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥𝑋))
101100ssrdv 3950 . . . . 5 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
102 cantnf.f . . . . 5 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
1031, 2, 3, 4, 55, 26, 101, 102cantnfp1 9617 . . . 4 (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
104103simprd 496 . . 3 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
10586oveq2d 7373 . . 3 (𝜑 → (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴o 𝑋) ·o 𝑌) +o 𝑍))
106104, 105, 443eqtrd 2780 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
1071, 2, 3cantnff 9610 . . . 4 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
108107ffnd 6669 . . 3 (𝜑 → (𝐴 CNF 𝐵) Fn 𝑆)
109103simpld 495 . . 3 (𝜑𝐹𝑆)
110 fnfvelrn 7031 . . 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 2839 1 (𝜑𝐶 ∈ ran (𝐴 CNF 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  c0 4282  ifcif 4486  cop 4592   cuni 4865   cint 4907   class class class wbr 5105  {copab 5167  cmpt 5188   E cep 5536  dom cdm 5633  ran crn 5634  Oncon0 6317  cio 6446   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920   supp csupp 8092  seqωcseqom 8393  1oc1o 8405  2oc2o 8406   +o coa 8409   ·o comu 8410  o coe 8411   finSupp cfsupp 9305  OrdIsocoi 9445   CNF ccnf 9597
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seqom 8394  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-oexp 8418  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-cnf 9598
This theorem is referenced by:  cantnflem4  9628
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