fuco2eld3 - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fuco2eld3		Structured version Visualization version GIF version

Theorem fuco2eld3 49346

Description: Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)

**Hypotheses**
Ref	Expression
fuco2eld.w	⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
fuco2eld2.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
fuco2eld2.s	⊢ Rel 𝑆
fuco2eld2.r	⊢ Rel 𝑅

**Assertion**
Ref	Expression
fuco2eld3	⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))𝑆(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))𝑅(2^nd ‘(2^nd ‘𝑈))))

**Proof of Theorem fuco2eld3**
Step	Hyp	Ref	Expression
1		fuco2eld2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
2		fuco2eld.w	. . . 4 ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
3		fuco2eld2.s	. . . 4 ⊢ Rel 𝑆
4		fuco2eld2.r	. . . 4 ⊢ Rel 𝑅
5	2, 1, 3, 4	fuco2eld2 49345	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
6	1, 5, 2	3eltr3d 2845	. 2 ⊢ (𝜑 → ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ (𝑆 × 𝑅))
7		fuco2el 49343	. 2 ⊢ (⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ (𝑆 × 𝑅) ↔ ((1^st ‘(1^st ‘𝑈))𝑆(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))𝑅(2^nd ‘(2^nd ‘𝑈))))
8	6, 7	sylib 218	1 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))𝑆(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))𝑅(2^nd ‘(2^nd ‘𝑈))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⟨cop 4582 class class class wbr 5091 × cxp 5614 Rel wrel 5621 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-iota 6437 df-fun 6483 df-fv 6489 df-1st 7921 df-2nd 7922

This theorem is referenced by: fucof21 49378

W3C validator