Theorem cantnflem1c 9608

Description: Lemma for cantnf 9614. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
oemapval.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
oemapval.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
oemapvali.r	⊢ (𝜑 → 𝐹𝑇𝐺)
oemapvali.x	⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
cantnflem1.o	⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))

Assertion

Ref	Expression
cantnflem1c	⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))

Distinct variable groups:   𝑢,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑢   𝑢,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑢,𝑥,𝑦,𝑧   𝐺,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑢,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑢,𝑥,𝑦,𝑧   𝑢,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑂(𝑐)   𝑋(𝑐)

Proof of Theorem cantnflem1c

Step	Hyp	Ref	Expression
1		cantnfs.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
2	1	ad3antrrr 731	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝐵 ∈ On)
3		simplr 769	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ 𝐵)
4		oemapval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
5		cantnfs.s	. . . . . . 7 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
6		cantnfs.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
7	5, 6, 1	cantnfs 9587	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
8	4, 7	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
9	8	simpld 494	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
10	9	ffnd 6671	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
11	10	ad3antrrr 731	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝐺 Fn 𝐵)
12		oemapval.t	. . . . . . 7 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
13		oemapval.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14		oemapvali.r	. . . . . . 7 ⊢ (𝜑 → 𝐹𝑇𝐺)
15		oemapvali.x	. . . . . . 7 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
16		cantnflem1.o	. . . . . . 7 ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))
17	5, 6, 1, 12, 13, 4, 14, 15, 16	cantnflem1b 9607	. . . . . 6 ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))
18	17	ad2antrr 727	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑋 ⊆ (𝑂‘𝑢))
19		simprr 773	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝑂‘𝑢) ∈ 𝑥)
20	5, 6, 1, 12, 13, 4, 14, 15	oemapvali 9605	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
21	20	simp1d 1143	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		onelon 6350	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
23	1, 21, 22	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ On)
24	23	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑋 ∈ On)
25		onss 7740	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
26	1, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ On)
27	26	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
28	27	ad4ant13 752	. . . . . 6 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ On)
29		ontr2 6373	. . . . . 6 ⊢ ((𝑋 ∈ On ∧ 𝑥 ∈ On) → ((𝑋 ⊆ (𝑂‘𝑢) ∧ (𝑂‘𝑢) ∈ 𝑥) → 𝑋 ∈ 𝑥))
30	24, 28, 29	syl2anc 585	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → ((𝑋 ⊆ (𝑂‘𝑢) ∧ (𝑂‘𝑢) ∈ 𝑥) → 𝑋 ∈ 𝑥))
31	18, 19, 30	mp2and 700	. . . 4 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑋 ∈ 𝑥)
32		eleq2w 2821	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑋 ∈ 𝑤 ↔ 𝑋 ∈ 𝑥))
33		fveq2 6842	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
34		fveq2 6842	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝐺‘𝑤) = (𝐺‘𝑥))
35	33, 34	eqeq12d 2753	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐺‘𝑤) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
36	32, 35	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑋 ∈ 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥))))
37	20	simp3d 1145	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
38	37	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
39	36, 38, 3	rspcdva 3579	. . . 4 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝑋 ∈ 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))
40	31, 39	mpd 15	. . 3 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝐹‘𝑥) = (𝐺‘𝑥))
41		simprl 771	. . 3 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝐹‘𝑥) ≠ ∅)
42	40, 41	eqnetrrd 3001	. 2 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → (𝐺‘𝑥) ≠ ∅)
43		fvn0elsupp 8132	. 2 ⊢ (((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)) → 𝑥 ∈ (𝐺 supp ∅))
44	2, 3, 11, 42, 43	syl22anc 839	1 ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865   class class class wbr 5100  {copab 5162   E cep 5531  ^◡ccnv 5631  dom cdm 5632  Oncon0 6325  suc csuc 6327   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   supp csupp 8112   finSupp cfsupp 9276  OrdIsocoi 9426   CNF ccnf 9582

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-seqom 8389  df-1o 8407  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-cnf 9583

This theorem is referenced by:  cantnflem1  9610