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Theorem ctiunctlemf 13004

Description: Lemma for ctiunct 13006. (Contributed by Jim Kingdon, 28-Oct-2023.)

**Hypotheses**
Ref	Expression
ctiunct.som	⊢ (𝜑 → 𝑆 ⊆ ω)
ctiunct.sdc	⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
ctiunct.f	⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
ctiunct.tom	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
ctiunct.g	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)
ctiunct.j	⊢ (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u	⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}
ctiunct.h	⊢ 𝐻 = (𝑛 ∈ 𝑈 ↦ (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))))

**Assertion**
Ref	Expression
ctiunctlemf	⊢ (𝜑 → 𝐻:𝑈⟶∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable groups: 𝐴,𝑛,𝑥 𝐵,𝑛 𝑥,𝐹,𝑧 𝑥,𝐽,𝑧 𝑧,𝑆 𝑧,𝑇 𝑈,𝑛 𝜑,𝑛,𝑥 𝑥,𝑧,𝑛 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝐵(𝑥,𝑧) 𝑆(𝑥,𝑛) 𝑇(𝑥,𝑛) 𝑈(𝑥,𝑧) 𝐹(𝑛) 𝐺(𝑥,𝑧,𝑛) 𝐻(𝑥,𝑧,𝑛) 𝐽(𝑛)

Proof of Theorem ctiunctlemf Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ctiunct.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
2	1	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → 𝐹:𝑆–onto→𝐴)
3		fof 5547	. . . . . . 7 ⊢ (𝐹:𝑆–onto→𝐴 → 𝐹:𝑆⟶𝐴)
4	2, 3	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → 𝐹:𝑆⟶𝐴)
5		ctiunct.som	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ω)
6	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → 𝑆 ⊆ ω)
7		ctiunct.sdc	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
8	7	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
9		ctiunct.tom	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)
10	9	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑈) ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)
11		ctiunct.tdc	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
12	11	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑈) ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
13		ctiunct.g	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)
14	13	adantlr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑈) ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)
15		ctiunct.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))
16	15	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → 𝐽:ω–1-1-onto→(ω × ω))
17		ctiunct.u	. . . . . . 7 ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}
18		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → 𝑛 ∈ 𝑈)
19	6, 8, 2, 10, 12, 14, 16, 17, 18	ctiunctlemu1st 13000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (1^st ‘(𝐽‘𝑛)) ∈ 𝑆)
20	4, 19	ffvelcdmd 5770	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (𝐹‘(1^st ‘(𝐽‘𝑛))) ∈ 𝐴)
21		fof 5547	. . . . . . . . . . 11 ⊢ (𝐺:𝑇–onto→𝐵 → 𝐺:𝑇⟶𝐵)
22	13, 21	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇⟶𝐵)
23	22	ralrimiva 2603	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐺:𝑇⟶𝐵)
24	23	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 𝐺:𝑇⟶𝐵)
25		rspsbc 3112	. . . . . . . 8 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑛))) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐺:𝑇⟶𝐵 → [(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥]𝐺:𝑇⟶𝐵))
26	20, 24, 25	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → [(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥]𝐺:𝑇⟶𝐵)
27		sbcfg 5471	. . . . . . . 8 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑛))) ∈ 𝐴 → ([(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥]𝐺:𝑇⟶𝐵 ↔ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺:⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝑇⟶⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵))
28	20, 27	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ([(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥]𝐺:𝑇⟶𝐵 ↔ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺:⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝑇⟶⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵))
29	26, 28	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺:⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝑇⟶⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵)
30	6, 8, 2, 10, 12, 14, 16, 17, 18	ctiunctlemu2nd 13001	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (2^nd ‘(𝐽‘𝑛)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝑇)
31	29, 30	ffvelcdmd 5770	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵)
32		csbeq1 3127	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(1^st ‘(𝐽‘𝑛))) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵)
33	32	eleq2d 2299	. . . . . 6 ⊢ (𝑦 = (𝐹‘(1^st ‘(𝐽‘𝑛))) → ((⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵))
34	33	rspcev 2907	. . . . 5 ⊢ (((𝐹‘(1^st ‘(𝐽‘𝑛))) ∈ 𝐴 ∧ (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵) → ∃𝑦 ∈ 𝐴 (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋𝑦 / 𝑥⦌𝐵)
35	20, 31, 34	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ∃𝑦 ∈ 𝐴 (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋𝑦 / 𝑥⦌𝐵)
36		eliun 3968	. . . 4 ⊢ ((⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ↔ ∃𝑦 ∈ 𝐴 (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋𝑦 / 𝑥⦌𝐵)
37	35, 36	sylibr 134	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵)
38		nfcv 2372	. . . 4 ⊢ Ⅎ𝑦𝐵
39		nfcsb1v 3157	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
40		csbeq1a 3133	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
41	38, 39, 40	cbviun 4001	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
42	37, 41	eleqtrrdi 2323	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
43		ctiunct.h	. 2 ⊢ 𝐻 = (𝑛 ∈ 𝑈 ↦ (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))))
44	42, 43	fmptd 5788	1 ⊢ (𝜑 → 𝐻:𝑈⟶∪ 𝑥 ∈ 𝐴 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 839 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 {crab 2512 [wsbc 3028 ⦋csb 3124 ⊆ wss 3197 ∪ ciun 3964 ↦ cmpt 4144 ωcom 4681 × cxp 4716 ⟶wf 5313 –onto→wfo 5315 –1-1-onto→wf1o 5316 ‘cfv 5317 1^st c1st 6282 2^nd c2nd 6283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-fo 5323 df-fv 5325

This theorem is referenced by: ctiunctlemfo 13005

W3C validator