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Theorem infxpenc2lem1 10016

Description: Lemma for infxpenc2 10019. (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 3246	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤)))
3	2	impr 453	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → ∃𝑤 ∈ (On ∖ 1_o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))
4		simpr 483	. . 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 7419	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (ω ↑_o 𝑥) = (ω ↑_o 𝑤))
7		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥)) = (𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))
8		ovex 7444	. . . . . . . . . 10 ⊢ (ω ↑_o 𝑤) ∈ V
9	6, 7, 8	fvmpt 6997	. . . . . . . . 9 ⊢ (𝑤 ∈ (On ∖ 1_o) → ((𝑥 ∈ (On ∖ 1_o) ↦ (ω ↑_o 𝑥))‘𝑤) = (ω ↑_o 𝑤))
10	9	ad2antrl 724	. . . . . . . 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 6839	. . . . . . . . . 10 ⊢ ((𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤) → (𝑛‘𝑏):𝑏–onto→(ω ↑_o 𝑤))
12	11	ad2antll 725	. . . . . . . . 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 6807	. . . . . . . . 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 2773	. . . . . . 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 7444	. . . . . . . . . . 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 9643	. . . . . . . . . . . . 13 ⊢ ω ∈ On
19		1onn 8641	. . . . . . . . . . . . 13 ⊢ 1_o ∈ ω
20		ondif2 8504	. . . . . . . . . . . . 13 ⊢ (ω ∈ (On ∖ 2_o) ↔ (ω ∈ On ∧ 1_o ∈ ω))
21	18, 19, 20	mpbir2an 707	. . . . . . . . . . . 12 ⊢ ω ∈ (On ∖ 2_o)
22		eldifi 4125	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (On ∖ 1_o) → 𝑥 ∈ On)
23	22	ad2antrl 724	. . . . . . . . . . . 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 4125	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (On ∖ 1_o) → 𝑦 ∈ On)
25	24	ad2antll 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))) → 𝑦 ∈ On)
26		oecan 8591	. . . . . . . . . . . 12 ⊢ ((ω ∈ (On ∖ 2_o) ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((ω ↑_o 𝑥) = (ω ↑_o 𝑦) ↔ 𝑥 = 𝑦))
27	21, 23, 25, 26	mp3an2i 1464	. . . . . . . . . . 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 411	. . . . . . . . . 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 8990	. . . . . . . . 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 6843	. . . . . . . . 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 767	. . . . . . . 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 7278	. . . . . . . 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 582	. . . . . . 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 2782	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) ∧ (𝑤 ∈ (On ∖ 1_o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑_o 𝑤))) → 𝑊 = 𝑤)
37	36	eleq1d 2816	. . . 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 7427	. . . . 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 6829	. . . 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 629	. . 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 3159	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 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 ↦ cmpt 5230 ^◡ccnv 5674 ran crn 5676 Oncon0 6363 –1-1→wf1 6539 –onto→wfo 6540 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7411 ωcom 7857 1_oc1o 8461 2_oc2o 8462 ↑_o coe 8467

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-oexp 8474

This theorem is referenced by: infxpenc2lem2 10017

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