dvhvscacbv - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvscacbv		Structured version Visualization version GIF version

Theorem dvhvscacbv 38236

Description: Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)

**Hypothesis**
Ref	Expression
dvhvscaval.s	⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)

**Assertion**
Ref	Expression
dvhvscacbv	⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Distinct variable groups: 𝑓,𝑠,𝑡,𝑔,𝐸 𝑇,𝑠,𝑓,𝑡,𝑔 Allowed substitution hints: · (𝑡,𝑓,𝑔,𝑠)

**Proof of Theorem dvhvscacbv**
Step	Hyp	Ref	Expression
1		dvhvscaval.s	. 2 ⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)
2		fveq1 6671	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑓)))
3		coeq1 5730	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑓)))
4	2, 3	opeq12d 4813	. . 3 ⊢ (𝑠 = 𝑡 → ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩)
5		2fveq3 6677	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡‘(1^st ‘𝑓)) = (𝑡‘(1^st ‘𝑔)))
6		fveq2 6672	. . . . 5 ⊢ (𝑓 = 𝑔 → (2^nd ‘𝑓) = (2^nd ‘𝑔))
7	6	coeq2d 5735	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑡 ∘ (2^nd ‘𝑓)) = (𝑡 ∘ (2^nd ‘𝑔)))
8	5, 7	opeq12d 4813	. . 3 ⊢ (𝑓 = 𝑔 → ⟨(𝑡‘(1^st ‘𝑓)), (𝑡 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
9	4, 8	cbvmpov 7251	. 2 ⊢ (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩) = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10	1, 9	eqtri 2846	1 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ⟨cop 4575 × cxp 5555 ∘ ccom 5561 ‘cfv 6357 ∈ cmpo 7160 1^st c1st 7689 2^nd c2nd 7690

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-co 5566 df-iota 6316 df-fv 6365 df-oprab 7162 df-mpo 7163

This theorem is referenced by: dvhvscaval 38237