fuco2eld3 - Mathbox for Zhi Wang

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Theorem fuco2eld3 49790

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 49789	. . 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 2850	. 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 49787	. 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 1542 ∈ wcel 2114 ⟨cop 4573 class class class wbr 5085 × cxp 5629 Rel wrel 5636 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6454 df-fun 6500 df-fv 6506 df-1st 7942 df-2nd 7943

This theorem is referenced by: fucof21 49822

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