Theorem tposidres 48755

Description: Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)

Hypotheses

Ref	Expression
tposidres.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tposidres.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
tposidres	⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

Proof of Theorem tposidres

Step	Hyp	Ref	Expression
1		ovtpos 8235	. . . 4 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (𝑋( I ↾ (𝐴 × 𝐵))𝑌)
2		df-ov 7403	. . . 4 ⊢ (𝑋( I ↾ (𝐴 × 𝐵))𝑌) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
3	1, 2	eqtri 2757	. . 3 ⊢ (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩)
4		tposidres.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5		tposidres.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	4, 5	opelxpd 5691	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
7	6	fvresd 6893	. . 3 ⊢ (𝜑 → (( I ↾ (𝐴 × 𝐵))‘⟨𝑋, 𝑌⟩) = ( I ‘⟨𝑋, 𝑌⟩))
8	3, 7	eqtrid 2781	. 2 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ( I ‘⟨𝑋, 𝑌⟩))
9		opex 5437	. . 3 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
10		fvi 6952	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∈ V → ( I ‘⟨𝑋, 𝑌⟩) = ⟨𝑋, 𝑌⟩)
11	9, 10	ax-mp 5	. 2 ⊢ ( I ‘⟨𝑋, 𝑌⟩) = ⟨𝑋, 𝑌⟩
12	8, 11	eqtrdi 2785	1 ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3457  ⟨cop 4605   I cid 5545   × cxp 5650   ↾ cres 5654  ‘cfv 6528  (class class class)co 7400  tpos ctpos 8219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-fv 6536  df-ov 7403  df-tpos 8220

This theorem is referenced by: (None)