Theorem infxpenc2lem1 9933

Description: Lemma for infxpenc2 9936. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
infxpenc2.1	⊢ (𝜑 → 𝐴 ∈ On)
infxpenc2.2	⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
infxpenc2.3	⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))

Assertion

Ref	Expression
infxpenc2lem1	⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))

Distinct variable groups:   𝑛,𝑏,𝑤,𝑥,𝐴   𝜑,𝑏,𝑤,𝑥   𝑤,𝑊,𝑥

Allowed substitution hints:   𝜑(𝑛)   𝑊(𝑛,𝑏)

Proof of Theorem infxpenc2lem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxpenc2.2	. . . 4 ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
2	1	r19.21bi 3229	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
3	2	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))
4		simpr 484	. . 3 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
5		infxpenc2.3	. . . . . 6 ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏))
6		oveq2 7368	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑o 𝑥) = (ω ↑o 𝑤))
7		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) = (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))
8		ovex 7393	. . . . . . . . . 10 ⊢ (ω ↑o 𝑤) ∈ V
9	6, 7, 8	fvmpt 6942	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1o) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = (ω ↑o 𝑤))
10	9	ad2antrl 729	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = (ω ↑o 𝑤))
11		f1ofo 6782	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤))
12	11	ad2antll 730	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤))
13		forn 6750	. . . . . . . . 9 ⊢ ((𝑛‘𝑏):𝑏–onto→(ω ↑o 𝑤) → ran (𝑛‘𝑏) = (ω ↑o 𝑤))
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → ran (𝑛‘𝑏) = (ω ↑o 𝑤))
15	10, 14	eqtr4d 2775	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = ran (𝑛‘𝑏))
16		ovex 7393	. . . . . . . . . . 11 ⊢ (ω ↑o 𝑥) ∈ V
17	16	2a1i 12	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑥 ∈ (On ∖ 1o) → (ω ↑o 𝑥) ∈ V))
18		omelon 9559	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19		1onn 8570	. . . . . . . . . . . . 13 ⊢ 1o ∈ ω
20		ondif2 8431	. . . . . . . . . . . . 13 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
21	18, 19, 20	mpbir2an 712	. . . . . . . . . . . 12 ⊢ ω ∈ (On ∖ 2o)
22		eldifi 4084	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1o) → 𝑥 ∈ On)
23	22	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1o) ∧ 𝑦 ∈ (On ∖ 1o))) → 𝑥 ∈ On)
24		eldifi 4084	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1o) → 𝑦 ∈ On)
25	24	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1o) ∧ 𝑦 ∈ (On ∖ 1o))) → 𝑦 ∈ On)
26		oecan 8519	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
27	21, 23, 25, 26	mp3an2i 1469	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1o) ∧ 𝑦 ∈ (On ∖ 1o))) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦))
28	27	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → ((𝑥 ∈ (On ∖ 1o) ∧ 𝑦 ∈ (On ∖ 1o)) → ((ω ↑o 𝑥) = (ω ↑o 𝑦) ↔ 𝑥 = 𝑦)))
29	17, 28	dom2lem 8933	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)):(On ∖ 1o)–1-1→V)
30		f1f1orn 6786	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)):(On ∖ 1o)–1-1→V → (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)):(On ∖ 1o)–1-1-onto→ran (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)):(On ∖ 1o)–1-1-onto→ran (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)))
32		simprl 771	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑤 ∈ (On ∖ 1o))
33		f1ocnvfv 7226	. . . . . . . 8 ⊢ (((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)):(On ∖ 1o)–1-1-onto→ran (𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥)) ∧ 𝑤 ∈ (On ∖ 1o)) → (((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = ran (𝑛‘𝑏) → (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏)) = 𝑤))
34	31, 32, 33	syl2anc 585	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘𝑤) = ran (𝑛‘𝑏) → (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏)) = 𝑤))
35	15, 34	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏)) = 𝑤)
36	5, 35	eqtrid 2784	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → 𝑊 = 𝑤)
37	36	eleq1d 2822	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑊 ∈ (On ∖ 1o) ↔ 𝑤 ∈ (On ∖ 1o)))
38	36	oveq2d 7376	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (ω ↑o 𝑊) = (ω ↑o 𝑤))
39	38	f1oeq3d 6772	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
40	37, 39	anbi12d 633	. . 3 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → ((𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)) ↔ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))))
41	4, 40	mpbird 257	. 2 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))
42	3, 41	rexlimddv 3144	1 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902   ↦ cmpt 5180  ^◡ccnv 5624  ran crn 5626  Oncon0 6318  –1-1→wf1 6490  –onto→wfo 6491  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  ωcom 7810  1_oc1o 8392  2_oc2o 8393   ↑_o coe 8398

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 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682  ax-inf2 9554

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 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-oexp 8405

This theorem is referenced by:  infxpenc2lem2  9934