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Theorem infxpenc2lem1 10017

Description: Lemma for infxpenc2 10020. (Contributed by Mario Carneiro, 30-May-2015.)

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

**Assertion**
Ref	Expression
infxpenc2lem1	⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1_o) ∧ (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 ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤)))
2	1	r19.21bi 3247	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤)))
3	2	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))
4		simpr 484	. . 3 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤)))
5		infxpenc2.3	. . . . . 6 ⊢ 𝑊 = (^◡(𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘ran (𝑛‘𝑏))
6		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑_o 𝑥) = (ω ↑_o 𝑤))
7		eqid 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)) = (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))
8		ovex 7445	. . . . . . . . . 10 ⊢ (ω ↑_o 𝑤) ∈ V
9	6, 7, 8	fvmpt 6999	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1_o) → ((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘𝑤) = (ω ↑_o 𝑤))
10	9	ad2antrl 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → ((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘𝑤) = (ω ↑_o 𝑤))
11		f1ofo 6841	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑_o 𝑤))
12	11	ad2antll 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑛‘𝑏):𝑏–onto→(ω ↑_o 𝑤))
13		forn 6809	. . . . . . . . 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 ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → ran (𝑛‘𝑏) = (ω ↑_o 𝑤))
15	10, 14	eqtr4d 2774	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → ((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘𝑤) = ran (𝑛‘𝑏))
16		ovex 7445	. . . . . . . . . . 11 ⊢ (ω ↑_o 𝑥) ∈ V
17	16	2a1i 12	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑥 ∈ (On ∖ 1_o) → (ω ↑_o 𝑥) ∈ V))
18		omelon 9644	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19		1onn 8642	. . . . . . . . . . . . 13 ⊢ 1_o ∈ ω
20		ondif2 8505	. . . . . . . . . . . . 13 ⊢ (ω ∈ (On ∖ 2_o) ↔ (ω ∈ On ∧ 1_o ∈ ω))
21	18, 19, 20	mpbir2an 708	. . . . . . . . . . . 12 ⊢ ω ∈ (On ∖ 2_o)
22		eldifi 4127	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1_o) → 𝑥 ∈ On)
23	22	ad2antrl 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1_o) ∧ 𝑦 ∈ (On ∖ 1_o))) → 𝑥 ∈ On)
24		eldifi 4127	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1_o) → 𝑦 ∈ On)
25	24	ad2antll 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1_o) ∧ 𝑦 ∈ (On ∖ 1_o))) → 𝑦 ∈ On)
26		oecan 8592	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2_o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑_o 𝑥) = (ω ↑_o 𝑦) ↔ 𝑥 = 𝑦))
27	21, 23, 25, 26	mp3an2i 1465	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) ∧ (𝑥 ∈ (On ∖ 1_o) ∧ 𝑦 ∈ (On ∖ 1_o))) → ((ω ↑_o 𝑥) = (ω ↑_o 𝑦) ↔ 𝑥 = 𝑦))
28	27	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → ((𝑥 ∈ (On ∖ 1_o) ∧ 𝑦 ∈ (On ∖ 1_o)) → ((ω ↑_o 𝑥) = (ω ↑_o 𝑦) ↔ 𝑥 = 𝑦)))
29	17, 28	dom2lem 8991	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)):(On ∖ 1_o)–1-1→V)
30		f1f1orn 6845	. . . . . . . . 9 ⊢ ((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)):(On ∖ 1_o)–1-1→V → (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)):(On ∖ 1_o)–1-1-onto→ran (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)):(On ∖ 1_o)–1-1-onto→ran (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)))
32		simprl 768	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → 𝑤 ∈ (On ∖ 1_o))
33		f1ocnvfv 7279	. . . . . . . 8 ⊢ (((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)):(On ∖ 1_o)–1-1-onto→ran (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)) ∧ 𝑤 ∈ (On ∖ 1_o)) → (((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘𝑤) = ran (𝑛‘𝑏) → (^◡(𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘ran (𝑛‘𝑏)) = 𝑤))
34	31, 32, 33	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘𝑤) = ran (𝑛‘𝑏) → (^◡(𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘ran (𝑛‘𝑏)) = 𝑤))
35	15, 34	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (^◡(𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘ran (𝑛‘𝑏)) = 𝑤)
36	5, 35	eqtrid 2783	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → 𝑊 = 𝑤)
37	36	eleq1d 2817	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑊 ∈ (On ∖ 1_o) ↔ 𝑤 ∈ (On ∖ 1_o)))
38	36	oveq2d 7428	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (ω ↑_o 𝑊) = (ω ↑_o 𝑤))
39	38	f1oeq3d 6831	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (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 630	. . 3 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → ((𝑊 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑊)) ↔ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))))
41	4, 40	mpbird 256	. 2 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → (𝑊 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑊)))
42	3, 41	rexlimddv 3160	1 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑊)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 ↦ cmpt 5232 ^◡ccnv 5676 ran crn 5678 Oncon0 6365 –1-1→wf1 6541 –onto→wfo 6542 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7412 ωcom 7858 1_oc1o 8462 2_oc2o 8463 ↑_o coe 8468

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7728 ax-inf2 9639

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-oexp 8475

This theorem is referenced by: infxpenc2lem2 10018

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