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Theorem ctiunctlemu1st 13303

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)}
ctiunctlem.n	⊢ (𝜑 → 𝑁 ∈ 𝑈)

**Assertion**
Ref	Expression
ctiunctlemu1st	⊢ (𝜑 → (1^st ‘(𝐽‘𝑁)) ∈ 𝑆)

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

**Proof of Theorem ctiunctlemu1st**
Step	Hyp	Ref	Expression
1		ctiunctlem.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑈)
2		2fveq3 5695	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (1^st ‘(𝐽‘𝑧)) = (1^st ‘(𝐽‘𝑁)))
3	2	eleq1d 2307	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ↔ (1^st ‘(𝐽‘𝑁)) ∈ 𝑆))
4		2fveq3 5695	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (2^nd ‘(𝐽‘𝑧)) = (2^nd ‘(𝐽‘𝑁)))
5	2	fveq2d 5694	. . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝐹‘(1^st ‘(𝐽‘𝑧))) = (𝐹‘(1^st ‘(𝐽‘𝑁))))
6	5	csbeq1d 3154	. . . . . . 7 ⊢ (𝑧 = 𝑁 → ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)
7	4, 6	eleq12d 2309	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 ↔ (2^nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇))
8	3, 7	anbi12d 477	. . . . 5 ⊢ (𝑧 = 𝑁 → (((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇) ↔ ((1^st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)))
9		ctiunct.u	. . . . 5 ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}
10	8, 9	elrab2 2985	. . . 4 ⊢ (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ ω ∧ ((1^st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)))
11	1, 10	sylib 122	. . 3 ⊢ (𝜑 → (𝑁 ∈ ω ∧ ((1^st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)))
12	11	simprd 114	. 2 ⊢ (𝜑 → ((1^st ‘(𝐽‘𝑁)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑁)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇))
13	12	simpld 112	1 ⊢ (𝜑 → (1^st ‘(𝐽‘𝑁)) ∈ 𝑆)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ∀wral 2528 {crab 2532 ⦋csb 3147 ⊆ wss 3220 ωcom 4732 × cxp 4767 –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-ext 2220

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-iota 5332 df-fv 5380

This theorem is referenced by: ctiunctlemf 13307

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