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Theorem ctiunctlemf 13307

Description: Lemma for ctiunct 13309. (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 5610	. . . . . . 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 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑈) ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)
11		ctiunct.tdc	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
12	11	adantlr 481	. . . . . . 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 481	. . . . . . 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 13303	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (1^st ‘(𝐽‘𝑛)) ∈ 𝑆)
20	4, 19	ffvelcdmd 5835	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (𝐹‘(1^st ‘(𝐽‘𝑛))) ∈ 𝐴)
21		fof 5610	. . . . . . . . . . 11 ⊢ (𝐺:𝑇–onto→𝐵 → 𝐺:𝑇⟶𝐵)
22	13, 21	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇⟶𝐵)
23	22	ralrimiva 2623	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐺:𝑇⟶𝐵)
24	23	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ∀𝑥 ∈ 𝐴 𝐺:𝑇⟶𝐵)
25		rspsbc 3135	. . . . . . . 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 5527	. . . . . . . 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 13304	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (2^nd ‘(𝐽‘𝑛)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝑇)
31	29, 30	ffvelcdmd 5835	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵)
32		csbeq1 3150	. . . . . . 7 ⊢ (𝑦 = (𝐹‘(1^st ‘(𝐽‘𝑛))) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐵)
33	32	eleq2d 2308	. . . . . 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 2929	. . . . 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 415	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑈) → ∃𝑦 ∈ 𝐴 (⦋(𝐹‘(1^st ‘(𝐽‘𝑛))) / 𝑥⦌𝐺‘(2^nd ‘(𝐽‘𝑛))) ∈ ⦋𝑦 / 𝑥⦌𝐵)
36		eliun 4011	. . . 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 2392	. . . 4 ⊢ Ⅎ𝑦𝐵
39		nfcsb1v 3180	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
40		csbeq1a 3156	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
41	38, 39, 40	cbviun 4044	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
42	37, 41	eleqtrrdi 2332	. 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 5853	1 ⊢ (𝜑 → 𝐻:𝑈⟶∪ 𝑥 ∈ 𝐴 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 [wsbc 3051 ⦋csb 3147 ⊆ wss 3220 ∪ ciun 4007 ↦ cmpt 4187 ωcom 4732 × cxp 4767 ⟶wf 5368 –onto→wfo 5370 –1-1-onto→wf1o 5371 ‘cfv 5372 1^st c1st 6362 2^nd c2nd 6363

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 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fo 5378 df-fv 5380

This theorem is referenced by: ctiunctlemfo 13308

W3C validator