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Theorem ctiunctlemu1st 13185

Description: Lemma for ctiunct 13191. (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 5675	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (1^st ‘(𝐽‘𝑧)) = (1^st ‘(𝐽‘𝑁)))
3	2	eleq1d 2301	. . . . . 6 ⊢ (𝑧 = 𝑁 → ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ↔ (1^st ‘(𝐽‘𝑁)) ∈ 𝑆))
4		2fveq3 5675	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (2^nd ‘(𝐽‘𝑧)) = (2^nd ‘(𝐽‘𝑁)))
5	2	fveq2d 5674	. . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝐹‘(1^st ‘(𝐽‘𝑧))) = (𝐹‘(1^st ‘(𝐽‘𝑁))))
6	5	csbeq1d 3145	. . . . . . 7 ⊢ (𝑧 = 𝑁 → ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑁))) / 𝑥⦌𝑇)
7	4, 6	eleq12d 2303	. . . . . 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 473	. . . . 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 2976	. . . 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 842 = wceq 1398 ∈ wcel 2203 ∀wral 2520 {crab 2524 ⦋csb 3138 ⊆ wss 3211 ωcom 4712 × cxp 4747 –onto→wfo 5350 –1-1-onto→wf1o 5351 ‘cfv 5352 1^st c1st 6332 2^nd c2nd 6333

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-ext 2214

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-rex 2526 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-un 3215 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-iota 5312 df-fv 5360

This theorem is referenced by: ctiunctlemf 13189

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